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Spacelike Singularities and Hidden Symmetries of Gravity

  • Marc Henneaux
  • Daniel Persson
  • Philippe Spindel
Open Access
Review Article

Abstract

We review the intimate connection between (super-)gravity close to a spacelike singularity (the “BKL-limit”) and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.

1 Introduction

It has been realized long ago that spacetime singularities are generic in classical general relativity [91]. However, their exact nature is still far from being well understood. Although it is expected that spacetime singularities will ultimately be resolved in a complete quantum theory of gravity, understanding their classical structure is likely to shed interesting light and insight into the nature of the mechanisms at play in the singularity resolution. Furthermore, analyzing general relativity close to such singularities also provides important information on the dynamics of gravity within the regime where it breaks down. Indeed, careful investigations of the field equations in this extreme regime has revealed interesting and unexpected symmetry properties of gravity.

In the late 1960’s, Belinskii, Khalatnikov and Lifshitz (“BKL”) [16] gave a general description of spacelike singularities in the context of the four-dimensional vacuum Einstein theory. They provided convincing evidence that the generic solution of the dynamical Einstein equations, in the vicinity of a spacelike singularity, exhibits the following remarkable properties:
  • The spatial points dynamically decouple, i.e., the partial differential equations governing the dynamics of the spatial metric asymptotically reduce, as one goes to the singularity, to ordinary differential equations with respect to time (one set of ordinary differential equations per spatial point).

  • The solution exhibits strong chaotic properties of the type investigated independently by Misner [137] and called “mixmaster behavior”. This chaotic behavior is best seen in the hyperbolic billiard reformulation of the dynamics due to Chitre [31] and Misner [138] (for pure gravity in four spacetime dimensions).

1.1 Cosmological billiards and hidden symmetries of gravity

This important work has opened the way to many further fruitful investigations in theoretical cosmology. Recently, a new — and somewhat unanticipated — development has occurred in the field, with the realisation that for the gravitational theories that have been studied most (pure gravity and supergravities in various spacetime dimensions) the dynamics of the gravitational field exhibits strong connections with Lorentzian Kac-Moody algebras, as discovered by Damour and Henneaux [45], suggesting that these might be “hidden” symmetries of the theory.

These connections appear for the cases at hand because in the BKL-limit, not only can the equations of motion be reformulated as dynamical equations for billiard motion in a region of hyperbolic space, but also this region possesses unique features: It is the fundamental Weyl chamber of some Kac-Moody algebra. The dynamical motion in the BKL-limit is then a succession of reflections in the walls bounding the fundamental Weyl chamber and defines “words” in the Weyl group of the Kac-Moody algebra.

Which billiard region of hyperbolic space actually emerges — and hence which Kac-Moody algebra is relevant — depends on the theory at hand, i.e., on the spacetime dimension, the menu of matter fields, and the dilaton couplings. The most celebrated case is eleven-dimensional super-gravity, for which the billiard region is the fundamental region of \({E_{10}} \equiv E_8^{+ +}\), one of the four hyperbolic Kac-Moody algebras of highest rank 10. The root lattice of E10 is furthermore one of the few even, Lorentzian, self-dual lattices — actually the only one in 10 dimensions — a fact that could play a key role in our ultimate understanding of M-theory.

Other gravitational theories lead to other billiards characterized by different algebras. These algebras are closely connected to the hidden duality groups that emerge upon dimensional reduction to three dimensions [41, 95].

That one can associate a regular billiard and an infinite discrete reflection group (Coxeter group) to spacelike singularities of a given gravitational theory in the BKL-limit is a robust fact (even though the BKL-limit itself is yet to be fully understood), which, in our opinion, will survive future developments. The mathematics necessary to appreciate the billiard structure and its connection to the duality groups in three dimensions involve hyperbolic Coxeter groups, Kac-Moody algebras and real forms of Lie algebras.

The appearance of infinite Coxeter groups related to Lorentzian Kac-Moody algebras has triggered fascinating conjectures on the existence of huge symmetry structures underlying gravity [47]. Similar conjectures based on different considerations had been made earlier in the pioneering works [113, 167]. The status of these conjectures, however, is still somewhat unclear since, in particular, it is not known how exactly the symmetry would act.

The main purpose of this article is to explain the emergence of infinite discrete reflection groups in gravity in a self-contained manner, including giving the detailed mathematical background needed to follow the discussion. We shall avoid, however, duplicating already existing reviews on BKL billiards.

Contrary to the main core of the review, devoted to an explanation of the billiard Weyl groups, which is indeed rather complete, we shall also discuss some paths that have been taken towards revealing the conjectured infinite-dimensional Kac-Moody symmetry. Our goal here will only be to give a flavor of some of the work that has been done along these lines, emphasizing its dynamical relevance. Because we feel that it would be premature to fully review this second subject, which is still in its infancy, we shall neither try to be exhaustive nor give detailed treatments.

1.2 Outline of the paper

Our article is organized as follows. In Section 2, we outline the key features of the BKL phenomenon, valid in any number of dimensions, and describe the billiard formulation which clearly displays these features. Since the derivation of these aspects have been already reviewed in [48], we give here only the results without proof. Next, for completeness, we briefly discuss the status of the BKL conjecture — assumed to be valid throughout our review.

In Sections 3 and 4, we develop the mathematical tools necessary for apprehending those aspects of Coxeter groups and Kac-Moody algebras that are needed in the BKL analysis. First, in Section 3, we provide a primer on Coxeter groups (which are the mathematical structures that make direct contact with the BKL billiards). We then move on to Kac-Moody algebras in Section 4, and we discuss, in particular, some prominent features of hyperbolic Kac-Moody algebras.

In Section 5 we then make use of these mathematical concepts to relate the BKL billiards to Lorentzian Kac-Moody algebras. We show that there is a simple connection between the relevant Kac-Moody algebra and the U-duality algebras that appear upon toroidal dimensional reduction to three dimensions, when these U-duality algebras are split real forms. The Kac-Moody algebra is then just the standard overextension of the U-duality algebra in question.

To understand the non-split case requires an understanding of real forms of finite-dimensional semi-simple Lie algebras. This mathematical material is developed in Section 6. Here, again, we have tried to be both rather complete and explicit through the use of many examples. We have followed a pedagogical approach privileging illustrative examples over complete proofs (these can be found in any case in the references given in the text). We explain the complementary Vogan and Tits-Satake approaches, where maximal compact and maximal noncompact Cartan subalgebras play the central roles, respectively. The concepts of restricted root systems and of the Iwasawa decomposition, central for understanding the emergence of the billiard, have been given particular attention. For completeness we provide tables listing all real forms of finite Lie algebras, both in terms of Vogan diagrams and in terms of Tits-Satake diagrams. In Section 7 we use these mathematical developments to relate the Kac-Moody billiards in the non-split case to the U-duality algebras appearing in three dimensions.

Up to (and including) Section 7, the developments present well-established results. With Section 8 we initiate a journey into more speculative territory. The presence of hyperbolic Weyl groups suggests that the corresponding infinite-dimensional Kac-Moody algebras might, in fact, be true underlying symmetries of the theory. How this conjectured symmetry should actually act on the physical fields is still unclear, however. We explore one approach in which the symmetry is realized nonlinearly on a (1 + 0)-dimensional sigma model based on \({{\mathcal E}_{10}}/{\mathcal K}({{\mathcal E}_{10}})\), which is the case relevant to eleven-dimensional supergravity. To this end, in Section 8 we introduce the concept of a level decomposition of some of the relevant Kac-Moody algebras in terms of finite regular subalgebras. This is necessary for studying the sigma model approach to the conjectured infinite-dimensional symmetries, a task undertaken in Section 9. We show that the sigma model for \({{\mathcal E}_{10}}/{\mathcal K}({{\mathcal E}_{10}})\) spectacularly reproduces important features of eleven-dimensional supergravity. However, we also point out important limitations of the approach, which probably does not constitute the final word on the subject.

In Section 10 we show that the interpretation of eleven-dimensional supergravity in terms of a manifestly \({{\mathcal E}_{10}}\)-invariant sigma model sheds interesting and useful light on certain cosmological solutions of the theory. These solutions were derived previously but without the Kac-Moody algebraic understanding. The sigma model approach also suggests a new method of uncovering novel solutions. Finally, in Section 11 we present a concluding discussion and some suggestions for future research.

2 The BKL Phenomenon

In this section, we explain the main ideas of the billiard description of the BKL behavior. Our approach is based on the billiard review [48], from which we adopt notations and conventions. We shall here only outline the logic and provide the final results. No attempt will be made to reproduce the (sometimes heuristic) arguments underlying the derivation.

2.1 The general action

We are interested in general theories describing Einstein gravity coupled to bosonic “matter” fields. The only known bosonic matter fields that consistently couple to gravity are p-form fields, so our collection of fields will contain, besides the metric, p-form fields, including scalar fields (p = 0). The action reads
$$S\left[ {{g_{\mu \nu}},\phi ,{A^{(p)}}} \right] = \int {{d^D}} x\;\sqrt {{- ^{(D)}}{\rm{g}}} \left[ {R - {\partial _\mu}\phi \;{\partial ^\mu}\phi - {1 \over 2}\sum\limits_p {{{{e^{{\lambda ^{(p)}}\phi}}} \over {(p + 1)!}}} F_{{\mu _1} \cdots {\mu _{p + 1}}}^{(p)}{F^{(p)\;{\mu _1} \cdots {\mu _{p + 1}}}}} \right] + {\prime\prime}{\rm{more}}{\prime\prime},$$
(2.1)
where we have chosen units such that 16πG = 1. The spacetime dimension is left unspecified. The Einstein metric g μν has Lorentzian signature (−, +, …, +) and is used to lower or raise the indices. Its determinant is(D)g, where the index D is used to avoid any confusion with the determinant of the spatial metric introduced below. We assume that among the scalars, there is only one dilaton1, denoted ϕ, whose kinetic term is normalized with weight 1 with respect to the Ricci scalar. The real parameter λ(p) measures the strength of the coupling to the dilaton. The other scalar fields, sometimes called axions, are denoted A(0) and have dilaton coupling λ(0) ≠ 0. The integer p ≥ 0 labels the various p-forms A(p) present in the theory, with field strengths F(p) = dA(p),
$$F_{{\mu _1} \cdots {\mu _{p + 1}}}^{(p)} = {\partial _{{\mu _1}}}A_{{\mu _2} \cdots {\mu _{p + 1}}}^{(p)} \pm p\;{\rm{permutations}}.$$
(2.2)
We assume the form degree p to be strictly smaller than D − 1, since a (D − 1)-form in D dimensions carries no local degree of freedom. Furthermore, if p = D − 2 the p-form is dual to a scalar and we impose also λ(D−2) ≠ 0.

The field strength, Equation (2.2), could be modified by additional coupling terms of Yang-Mills or Chapline-Manton type [20, 29] (e.g., F C = dC(2)C(0)dB(2) for two 2-forms C(2) and B(2) and a 0-form C(0), as it occurs in ten-dimensional type IIB supergravity), but we include these additional contributions to the action in “more”. Similarly, “more” might contain Chern-Simons terms, as in the action for eleven-dimensional supergravity [38].

We shall at this stage consider arbitrary dilaton couplings and menus of p-forms. The billiard derivation given below remains valid no matter what these are; all theories described by the general action Equation (2.1) lead to the billiard picture. However, it is only for particular p-form menus, spacetime dimensions and dilaton couplings that the billiard region is regular and associated with a Kac-Moody algebra. This will be discussed in Section 5. Note that the action, Equation (2.1), contains as particular cases the bosonic sectors of all known supergravity theories.

2.2 Hamiltonian description

We assume that there is a spacelike singularity at a finite distance in proper time. We adopt a spacetime slicing adapted to the singularity, which “occurs” on a slice of constant time. We build the slicing from the singularity by taking pseudo-Gaussian coordinates defined by \(N = \sqrt g\) and N i = 0, where N is the lapse and N i is the shift [48]. Here, g = det(g ij ). Thus, in some spacetime patch, the metric reads2
$$d{s^2} = - {\rm{g}}{(d{x^0})^2} + {{\rm{g}}_{ij}}({x^0},{x^i})\;d{x^i}\;d{x^j},$$
(2.3)
where the local volume g collapses at each spatial point as x0 → +∞, in such a way that the proper time \(dT = - \sqrt g d{x^0}\) remains finite (and tends conventionally to 0+). Here we have assumed the singularity to occur in the past, as in the original BKL analysis, but a similar discussion holds for future spacelike singularities.

2.2.1 Action in canonical form

In the Hamiltonian description of the dynamics, the canonical variables are the spatial metric components g ij , the dilaton ϕ, the spatial p-form components \(A_{{m_1}\, \cdots {m_p}}^{(p)}\) and their respective conjugate momenta π ij , πϕ and\(\pi _{(p)}^{{m_1}\, \cdots {m_p}}\). The Hamiltonian action in the pseudo-Gaussian gauge is given by
$$S\left[ {{{\rm{g}}_{ij}},{\pi ^{ij}},\phi ,{\pi _\phi},A_{{m_1} \cdots {m_p}}^{(p)},\pi _{(p)}^{{m_1} \cdots {m_p}}} \right]\; = \;\int d{x^0}\left[ {\int {{d^d}} x\left({{\pi ^{ij}}{{\dot g}_{ij}} + {\pi _\phi}\dot \phi + \sum\limits_p {\pi _{(p)}^{{m_1} \cdots {m_p}}} \dot A_{{m_1} \cdots {m_p}}^{(p)}} \right) - H} \right],$$
(2.4)
where the Hamiltonian is
$$\begin{array}{*{20}c} {H = \int {{d^d}} x\;{\mathcal H},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{\mathcal H} = K\prime + V\prime ,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;} \\ {K\prime = {\pi ^{ij}}{\pi _{ij}} - {1 \over {d - 1}}{{(\pi _i^i)}^2} + {1 \over 4}{{({\pi _\phi})}^2} + \sum\limits_p {{{(p!){e^{- {\lambda ^{(p)}}\phi}}} \over 2}} \;\pi _{(p)}^{{m_1} \cdots {m_p}}{\pi _{(p)\;{m_1} \cdots {m_p}}},\;} \\ {V\prime = - Rg + {g^{ij}}g{\partial _i}\phi {\partial _j}\phi + \sum\limits_p {{{{e^{{\lambda ^{(p)}}\phi}}} \over {2(p + 1)!}}g\;F_{{m_1} \cdots {m_{p + 1}}}^{(p)}{F^{(p){m_1} \cdots {m_{p + 1}}}}} .\quad \quad \quad \;} \\ \end{array}$$
(2.5)
In addition to imposing the coordinate conditions \(N = \sqrt g\) and N i = 0, we have also set the temporal components of the p-forms equal to zero (“temporal gauge”).
The dynamical equations of motion are obtained by varying the above action w.r.t. the canonical variables. Moreover, there are constraints on the dynamical variables, which are
$$\begin{array}{*{20}c} {{\mathcal{H}} = 0} & {(``{\rm Hamiltonian}\;\;{\rm constraint}"),\quad \quad \quad \quad} \\ {{{\mathcal {H}}_i} = 0} & {\;(``{\rm{momentum}}\;\;\;{\rm{constraint}}"),\quad \quad \quad \quad \quad} \\ {\varphi _{(p)}^{{m_1} \cdots {m_{p - 1}}} = 0\;\;\;\;\;\;\;\;\;\;} & {(``{\rm{Gauss}}\;\;{\rm{law}}"\;\; {\rm {for}}\;\;{\rm{each}}\;\;p-{\rm{form}},\;\;p > 0){.}} \\ \end{array}$$
(2.6)
Here we have set
$$\begin{array}{*{20}c} {{{\mathcal {H}}_i} = - 2\pi _{i\;\;\vert j}^j + {\pi _\phi}{\partial _i}\phi + \sum\limits_p {\pi _{(p)}^{{m_1} \cdots {m_p}}} F_{i{m_1} \cdots {m_p}}^{(p)},} \\ \varphi _{(p)}^{{m}_{1} \cdots {m}_{p - 1}} = - p\;\pi _{(p)}^{{m}_{1} \cdots {m}_{p - 1}{m}_{p}}{\vert {m}_{p},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(2.7)
where the subscript |m p denotes the spatially covariant derivative. These constraints are preserved by the dynamical evolution and need to be imposed only at one “initial” time, say at x0 = 0.

2.2.2 Iwasawa change of variables

In order to study the dynamical behavior of the fields as x0 → ∞ (g → 0) and to exhibit the billiard picture, it is particularly convenient to perform the Iwasawa decomposition of the spatial metric. Let g(x0,x i ) be the matrix with entries g ij (x0,x i ). We set
$$g = {\mathcal N}^{T}\;{\mathcal A}^{2}\;\;{\mathcal N},$$
(2.8)
where \({\mathcal N}\) is an upper triangular matrix with 1’s on the diagonal (\({{\mathcal N}_{ii}} = 1,\,{{\mathcal N}_{ij}} = 0\,{\rm{for}} \, i{\rm{>}}j\)) and A is a diagonal matrix with positive elements, which we parametrize as
$${\mathcal {A}} = \exp (- \beta),\qquad \beta = {\rm{diag}}({\beta ^1},{\beta ^2}, \cdots ,{\beta ^d}){.}$$
(2.9)
Both \(\mathcal N\) and \(\mathcal A\) depend on the spacetime coordinates. The spatial metric 2 becomes
$$d{\sigma ^2} = {{\rm{g}}_{ij}}\;d{x^i}\;d{x^j} = \sum\limits_{k = 1}^d {{e^{(- 2{\beta ^k})}}} {({\omega ^k})^2}$$
(2.10)
with
$${\omega ^k} = \sum\limits_i {{{\mathcal {N}}_{k\;i}}} \;d{x^i}.$$
(2.11)
The variables β i of the Iwasawa decomposition give the (logarithmic) scale factors in the new, orthogonal, basis. The variables \({{\mathcal N}_{ij}}\) characterize the change of basis that diagonalizes the metric and hence they parametrize the off-diagonal components of the original g ij .
We extend the transformation Equation (2.8) in configuration space to a canonical transformation in phase space through the formula
$${\pi ^{ij}}d{{\rm{g}}_{ij}} = {\pi ^i}d{\beta _i} + \sum\limits_{i < j} {{P_{ij}}} \;d{{\mathcal {N}}_{ij}}.$$
(2.12)
Since the scale factors and the off-diagonal variables play very distinct roles in the asymptotic behavior, we split off the Hamiltonian as a sum of a kinetic term for the scale factors (including the dilaton),
$$K = {1 \over 4}\left[ {\sum\limits_{i = 1}^d {\pi _i^2} - {1 \over {d - 1}}{{\left({\sum\limits_{i = 1}^d {{\pi _i}}} \right)}^2} + \pi _\phi ^2} \right],$$
(2.13)
plus the rest, denoted by V, which will act as a potential for the scale factors. The Hamiltonian then becomes
$$\begin{array}{*{20}c} {{\mathcal {H}} = K + V,\quad \quad \quad \quad \quad} \\ {V = {V_S} + {V_G} + \sum\limits_p {{V_p}} + {V_\phi},} \\ {\quad \quad \quad {V_S} = {1 \over 2}\sum\limits_{i < j} {{e^{- 2({\beta ^j} - {\beta ^i})}}} {{\left(\sum\limits_m {{P_{im}}} {{\mathcal N}_{jm}}\right)}^2},} \\ {{V_G} = - {R_g},\quad \quad \quad \quad \quad \quad} \\ {{V_{(p)}} = V_{(p)}^{{\rm{el}}} + V_{(p)}^{{\rm{magn}}},\quad \quad \quad \quad} \\ {\quad V_{(p)}^{{\rm{el}}} = {{p!{e^{- {\lambda ^{(p)}}\phi}}} \over 2}\;\pi _{(p)}^{{m_1} \cdots {m_p}}{\pi _{(p)\;{m_1} \cdots {m_p}}},} \\ {\quad V_{(p)}^{{\rm{magn}}} = {{{e^{{\lambda ^{(p)}}\phi}}} \over {2\;(p + 1)!}}\;{\rm{g}}\;F_{{m_1} \cdots {m_{p + 1}}}^{(p)}{F^{(p)\;{m_1} \cdots {m_{p + 1}}}},} \\ {{V_\phi} = {{\rm{g}}^{ij}}g\;{\partial _i}\phi \;{\partial _j}\phi .\quad \quad \quad \quad} \\ \end{array}$$
(2.14)
The kinetic term K is quadratic in the momenta conjugate to the scale factors and defines the inverse of a metric in the space of the scale factors. Explicitly, this metric reads
$$\sum\limits_i {{{(d{\beta ^i})}^2}} - {(\sum d {\beta ^i})^2} + {(d\phi)^2}.$$
(2.15)
Since the metric coefficients do not depend on the scale factors, that metric in the space of scale factors is flat, and, moreover, it is of Lorentzian signature. A conformal transformation where all scale factors are scaled by the same number (β i β i + ϵ) defines a timelike direction. It will be convenient in the following to collectively denote all the scale factors (the β i ’s and the dilaton ϕ) as β μ , i.e., (β μ ) = (β i ϕ).
The analysis is further simplified if we take for new p-form variables the components of the p-forms in the Iwasawa basis of the ω k ,
$${\mathcal A}_{{i_1} \cdots {i_p}}^{(p)} = \sum\limits_{{m_1}, \cdots ,{m_p}} {({{\mathcal {N}}^{- 1}})_{{m_1}{i_1}}} \cdots {({{\mathcal {N}}^{- 1}})_{{m_p}{i_p}}}{A_{(p){m_1} \cdots {m_p}}},$$
(2.16)
and again extend this configuration space transformation to a point canonical transformation in phase space,
$$\left({{{\mathcal{N}}_{ij}},{P_{ij}},A_{{m_1} \cdots {m_p}}^{(p)},\pi _{(p)}^{{m_1} \cdots {m_p}}} \right)\quad \rightarrow \quad \left({{{\mathcal{N}}_{ij}},{{P\prime}_{ij}},{\mathcal A}_{{m_1} \cdots {m_p}}^{(p)},{\mathcal {E}}_{(p)}^{{i_1} \cdots {i_p}}} \right),$$
(2.17)
using the formula ∑ p dq = ∑p′ dq′, which reads
$$\sum\limits_{i < j} {{P_{ij}}} {{\mathcal {\dot N}}_{ij}} + \sum\limits_p {\pi _{(p)}^{{m_1} \cdots {m_p}}} \dot {\mathcal A}_{{m_1} \cdots {m_p}}^{(p)} = \sum\limits_{i < j} {{{P\prime}_{ij}}} {{\mathcal {\dot N}}_{ij}} + \sum\limits_p {{\mathcal {E}}_{(p)}^{{i_1} \cdots {i_p}}} \dot A_{{m_1} \cdots {m_p}}^{(p)}.$$
(2.18)
Note that the scale factor variables are unaffected, while the momenta P ij conjugate to \({{\mathcal N}_{ij}}\) get redefined by terms involving \({\mathcal E}\), \({\mathcal N}\) and \({\mathcal A}\) since the components \(A_{{m_1}\, \cdots {m_p}}^{(p)}\) of the p-forms in the Iwasawa basis involve the \({\mathcal N}\)’s. On the other hand, the new p-form momenta, i.e., the components of the electric field \(\pi _{(p)}^{{m_{1\, \cdots}}{m_p}}\) in the basis {ω k } are simply given by
$${\mathcal E}_{(p)}^{{i_1} \cdots {i_p}} = \sum\limits_{{m_1}, \cdots ,{m_p}} {{{\mathcal N}_{{i_1}{m_1}}}} {{\mathcal N}_{{i_2}{m_2}}} \cdots {{\mathcal N}_{{i_p}{m_p}}}\pi _{(p)}^{{m_1} \cdots {m_p}}.$$
(2.19)
In terms of the new variables, the electromagnetic potentials become
$$\begin{array}{*{20}c} {V_{(p)}^{{\rm{el}}} = {{p!} \over 2}\sum\limits_{{i_1},{i_2}, \cdots ,{i_p}} {e^{- 2{e_{{i_1} \cdots {i_p}}}(\beta)}}{{(\mathcal{E}_{(p)}^{{i_1} \cdots {i_p}})}^2},} \\ {V_{(p)}^{{\rm{magn}}} = {1 \over {2\;(p + 1)!}}\sum\limits_{{i_1},{i_2}, \cdots ,{i_{p + 1}}} {e^{- 2{m_{{i_1} \cdots {i_{p + 1}}}}(\beta)}}{{({\mathcal{F}_{(p)\;{i_1} \cdots {i_{p + 1}}}})}^2}.} \\ \end{array}$$
(2.20)
Here, \({e_{{i_1} \ldots {i_p}}}(\beta)\) are the electric linear forms
$${e_{{i_1} \cdots {i_p}}}(\beta) = {\beta ^{{i_1}}} + \cdots + {\beta ^{{i_p}}} + {{{\lambda ^{(p)}}} \over 2}\phi$$
(2.21)
(the indices i j are all distinct because \({\mathcal E}_{(p)}^{{i_1} \cdots {i_p}}\) is completely antisymmetric) while \({{\mathcal F}_{(p)\,{i_1} \cdots {i_{p + 1}}}}\) are the components of the magnetic field \({F_{(p){m_1}\, \cdots {m_{p + 1}}}}\) in the basis {ω k },
$${\mathcal{F}_{(p)\;{i_1} \cdots {i_{p + 1}}}} = \sum\limits_{{m_1}, \cdots ,{m_{p + 1}}} {({\mathcal{N}^{- 1}})_{{m_1}{i_1}}} \cdots {({\mathcal{N}^{- 1}})_{{m_{p + 1}}{i_{p + 1}}}}{F_{(p){m_1} \cdots {m_{p + 1}}}},$$
(2.22)
and \({m_{{i_1} \cdots {i_{p + 1}}}}(\beta)\) are the magnetic linear forms
$${m_{{i_1} \cdots {i_{p + 1}}}}(\beta) = \sum\limits_{j \notin \{{i_1},{i_2}, \cdots {i_{p + 1}}\}} {{\beta ^j}} - {{{\lambda ^{(p)}}} \over 2}\phi .$$
(2.23)
One sometimes rewrites \({m_{{i_1} \cdots {i_{p + 1}}}}(\beta)\) as \({b_{{i_{p + 2 \cdots {i_d}}}}}(\beta)\), where {ip+2, ip+3, …, i d } is the set complementary to {i1, i2, … ip+1}, e.g.,
$${b_{1\;2\; \cdots \;d - p - 1}}(\beta) = {\beta ^1} + \cdots + {\beta ^{d - p - 1}} - {{{\lambda ^{(p)}}} \over 2}\phi = {m_{d - p\; \cdots \;d}}.$$
(2.24)
The exterior derivative \({\mathcal F}\) of \({\mathcal A}\) in the non-holonomic frame {ω k } involves of course the structure coefficients \({C^i}_{jk}\) in that frame, i.e.,
$${\mathcal{F}_{(p)\;{i_1} \cdots {i_{p + 1}}}} = {\partial_{\left[ {{i_1}} \right.}}{\mathcal{A}_{\left. {{i_2} \cdots {i_{p + 1}}} \right]}} + ``{C}\mathcal{A}"-{\rm{terms}},$$
(2.25)
where
$${\partial _{{i_1}}} \equiv \sum\limits_{{m_1}} {{{({\mathcal{N}^{- 1}})}_{{m_1}{i_1}}}} (\partial /\partial {x^{{m_1}}})$$
(2.26)
is here the frame derivative. Similarly, the potential V ϕ reads
$${V_\phi} = \sum\limits_i {{e^{- 2{{\bar m}_i}(\beta)}}} {({\mathcal{F}_i})^2},$$
(2.27)
where \({{\mathcal F}_i}\) is
$${\mathcal{F}_i} = {({\mathcal{N}^{- 1}})_{ji}}{\partial _j}\phi$$
(2.28)
and
$${\bar m_i}(\beta) = \sum\limits_{j\not = i} {{\beta ^j}} .$$
(2.29)

2.3 Decoupling of spatial points close to a spacelike singularity

So far we have only redefined the variables without making any approximation. We now start the discussion of the BKL-limit, which investigates the leading behavior of the fields as x0 (g → 0). Although the more recent “derivations” of the BKL-limit treat both elements at once [43, 44, 45, 48], it appears useful — especially for rigorous justifications — to separate two aspects of the BKL conjecture3.

The first aspect is that the spatial points decouple in the limit x0 → ∞, in the sense that one can replace the Hamiltonian by an effective “ultralocal” Hamiltonian HUL involving no spatial gradients and hence leading at each point to a set of dynamical equations that are ordinary differential equations with respect to time. The ultralocal effective Hamiltonian has a form similar to that of the Hamiltonian governing certain spatially homogeneous cosmological models, as we shall explain in this section.

The second aspect of the BKL-limit is to take the sharp wall limit of the ultralocal Hamiltonian. This leads directly to the billiard description, as will be discussed in Section 2.4.

2.3.1 Spatially homogeneous models

In spatially homogeneous models, the fields depend only on time in invariant frames, e.g., for the metric
$$d{s^2} = {{\rm{g}}_{ij}}({x^0}){\psi ^i}{\psi ^j},$$
(2.30)
where the invariant forms fulfill
$$d{\psi ^i} = - {1 \over 2}{f^i}_{jk}{\psi ^j} \wedge {\psi ^k}.$$
Here, the \({f^i}_{jk}\) are the structure constants of the spatial homogeneity group. Similarly, for a 1-form and a 2-form,
$${A^{(1)}} = {A_i}({x^0}){\psi ^i},\qquad {A^{(2)}} = {1 \over 2}{A_{ij}}({x^0}){\psi ^i} \wedge {\psi ^j},\qquad {\rm{etc}}{.}$$
(2.31)
The Hamiltonian constraint yielding the field equations in the spatially homogeneous context4 is obtained by substituting the form of the fields in the general Hamiltonian constraint and contains, of course, no explicit spatial gradients since the fields are homogeneous. Note, however, that the structure constants \({f^i}_{jk}\) contain implicit spatial gradients. The Hamiltonian can now be decomposed as before and reads
$$\begin{array}{*{20}c} {{\mathcal{H}^{{\rm{UL}}}} = K + {V^{{\rm{UL}}}},\quad \quad \quad \quad \quad \quad \quad \quad \;} \\ {{V^{{\rm{UL}}}} = {V_S} + V_G^{{\rm{UL}}} + \sum\limits_p {\left({V_{(p)}^{{\rm{el}}} + V_{(p)}^{{\rm{UL}},{\rm{magn}}}} \right)} ,} \\ \end{array}$$
(2.32)
where K, V S and \(V_{(p)}^{{\rm{el}}}\) which do not involve spatial gradients, are unchanged and where V ϕ disappears since \({\partial _i}\phi = 0\). The potential V G is given by [61]
$${V_G} \equiv - {\rm{g}}R = {1 \over 4}\sum\limits_{i\not = j,i\not = k,j\not = k} {{e^{- 2{\alpha _{ijk}}(\beta)}}} {({C^i}_{jk})^2} + {1 \over 2}\sum\limits_j {{e^{- 2{{\bar m}_j}(\beta)}}} \left({{C^i}_{jk}\;{C^k}_{ji} + ``\mathrm{more}"} \right),$$
(2.33)
where the linear forms α ijk (β) (with i, j, k distinct) read
$${\alpha _{ijk}}(\beta) = 2{\beta ^i} + \sum\limits_{m\;:\;m\not = i,m\not = j,m\not = k} {{\beta ^m}} ,$$
(2.34)
and where “more” stands for the terms in the first sum that arise upon taking i = j or i = k. The structure constants in the Iwasawa frame (with respect to the coframe in Equation (2.30)) are related to the structure constants \({f^i}_{jk}\) through
$${C^i}_{jk} = \sum\limits_{i\prime ,j\prime ,k\prime} {{f^{i\prime}}_{j\prime k\prime}} \mathcal{N}_{ii\prime}^{- 1}{\mathcal{N}_{jj\prime}}{\mathcal{N}_{kk\prime}}$$
(2.35)
and depend therefore on the dynamical variables. Similarly, the potential \(V_{(p)}^{{\rm{magn}}}\) becomes
$$V_{(p)}^{{\rm{magn}}} = {1 \over {2\;(p + 1)!}}\sum\limits_{{i_1},{i_2}, \cdots ,{i_{p + 1}}} {{e^{- 2{m_{{i_1} \cdots {i_{p + 1}}}}(\beta)}}} {(\mathcal{F}_{(p)\;{i_1} \cdots {i_{p + 1}}}^h)^2},$$
(2.36)
where the field strengths \({\mathcal F}_{(p)\,{i_1} \cdots {i_{p + 1}}}^h\) reduce to the “AC” terms in dA and depend on the potentials and the off-diagonal Iwasawa variables.

2.3.2 The ultralocal Hamiltonian

Let us now come back to the general, inhomogeneous case and express the dynamics in the frame {dx0, ψ i } where the ψ i ’s form a “generic” non-holonomic frame in space,
$$d{\psi ^i} = - {1 \over 2}{f^i}_{jk}({x^m})\;{\psi ^j} \wedge {\psi ^k}.$$
(2.37)
Here the \({f^i}_{jk}\)’s are in general space-dependent. In the non-holonomic frame, the exact Hamiltonian takes the form
$$\mathcal{H} = {\mathcal{H}^{{\rm{UL}}}} + {\mathcal{H}^{{\rm{gradient}}}},$$
(2.38)
where the ultralocal part \({{\mathcal H}^{{\rm{UL}}}}\) is given by Equations (2.32) and (2.33) with the relevant \({f^i}_{jk}\)’s, and where \({{\mathcal H}^{{\rm{gradient}}}}\) involves the spatial gradients of \({f^i}_{jk},\,{\beta ^m},\,\phi\) and \({{\mathcal N}_{ij}}\).
The first part of the BKL conjecture states that one can drop \({{\mathcal H}^{{\rm{gradient}}}}\) asymptotically; namely, the dynamics of a generic solution of the Einstein-p-form-dilaton equations (not necessarily spatially homogeneous) is asymptotically determined, as one goes to the spatial singularity, by the ultralocal Hamiltonian
$${H^{{\rm{UL}}}} = \int {{d^d}} x\;{\mathcal{H}^{{\rm{UL}}}},$$
(2.39)
provided that the phase space constants \({f^i}_{jk}({x^m}) = - {f^i}_{kj}({x^m})\) are such that all exponentials in the above potentials do appear. In other words, the f’s must be chosen such that none of the coefficients of the exponentials, which involve f and the fields, identically vanishes — as would be the case, for example, if \({f^i}_{jk} = 0\) since then the potentials V G and \(V_{(p)}^{{\rm{magn}}}\) are equal to zero. This is always possible because the \({f^i}_{jk}\), even though independent of the dynamical variables, may in fact depend on x and so are not required to fulfill relations “f f = 0” analogous to the Bianchi identity since one has instead “∂f + f f = 0”.
2.3.2.1 Comments
  1. 1.

    As we shall see, the conditions on the f’s (that all exponentials in the potential should be present) can be considerably weakened. It is necessary that only the relevant exponentials (in the sense defined in Section 2.4) be present. Thus, one can correctly capture the asymptotic BKL behavior of a generic solution with fewer exponentials. In the case of eleven-dimensional supergravity the spatial curvature is asymptotically negligible with respect to the electromagnetic terms and one can in fact take a holonomic frame for which \({f^i}_{jk} = 0\) (and hence also \({C^i}_{jk} = 0\).

     
  2. 2.

    The actual values of the \({f^i}_{jk}\) (provided they fulfill the criterion given above or rather its weaker form just mentioned) turn out to be irrelevant in the BKL-limit because they can be absorbed through redefinitions. This is for instance why the Bianchi VIII and IX models, even though they correspond to different groups, can both be used to describe the BKL behavior in four spacetime dimensions.

     

2.4 Dynamics as a billiard in hyperbolic space

The second step in the BKL-limit is to take the sharp wall limit of the potentials.5 This leads to the billiard picture. It is crucial here that the coefficients in front of the dominant walls are all positive. Again, just as for the first step, this limit has not been fully justified. Only heuristic, albeit convincing, arguments have been put forward.

The idea is that as one goes to the singularity, the exponential potentials get sharper and sharper and can be replaced in the limit by the corresponding Θ∞-function, denoted for short Θ and defined by Θ(x) = 0 for x < 0 and Θ(x) = +∞ for x > 0. Taking into account the facts that aΘ(x) = Θ(x) for all a > 0, as well as that some walls can be neglected, one finds that the Hamiltonian becomes in the sharp wall limit
$$H = \int {{d^d}} x\;{\mathcal{H}^{{\rm{sharp}}}},$$
(2.40)
with
$$\begin{array}{*{20}c} {{\mathcal{H}^{{\rm{sharp}}}} = K + \sum\limits_{i < j} \Theta \left({- 2{s_{ji}}(\beta)} \right) + \sum\limits_{i\not = j,i\not = k,j\not = k} \Theta (- 2{\alpha _{ijk}}(\beta))\quad \quad \quad \quad \quad \quad} \\ {+ \sum\limits_{{i_1} < {i_2} < \cdots < {i_p}} \Theta (- 2{e_{{i_1} \cdots {i_p}}}(\beta)) + \sum\limits_{{i_1} < {i_2} < \cdots < {i_{p + 1}}} \Theta (- 2{m_{{i_1} \cdots {i_{p + 1}}}}(\beta)),} \\ \end{array}$$
(2.41)
where s ji (β) = β j β i . See [48] for more information.
The description of the motion of the scale factors (at each spatial point) is easy to give in that limit. Because the potential walls are infinite (and positive), the motion is constrained to the region where the arguments of all Θ-functions are negative, i.e., to
$${s_{ji}}(\beta) \geq 0\;(i < j),\qquad {\alpha _{ijk}}(\beta) \geq 0,\qquad {e_{{i_1} \cdots {i_p}}}(\beta) \geq 0,\qquad {m_{{i_1} \cdots {i_{p + 1}}}}(\beta) \geq 0.$$
(2.42)
In that region, the motion is governed by the kinetic term K, i.e., is a geodesic for the metric in the space of the scale factors. Since that metric is flat, this is a straight line. In addition, the constraint \({\mathcal H} = 0\), which reduces to K=0 away from the potential walls, forces the straight line to be null. We shall assume that the time orientation in the space of the scale factors is such that the straight line is future-oriented (g → 0 in the future).
It is easy to check that all the walls appearing in Equation (2.41), collectively denoted \({F_A}(\beta) \equiv {F_{A\mu}}{\beta ^\mu} = 0\), are timelike hyperplanes. This is because the squared norms of all the F a ’s are positive,
$$({F_A}\vert {F_A}) = \sum\limits_i {{{\left({{{\partial {F_A}} \over {\partial {\beta ^i}}}} \right)}^2}} - {1 \over {d - 1}}{\left({\sum\limits_i {{{\partial {F_A}} \over {\partial {\beta ^i}}}}} \right)^2} + {\left({{{\partial {F_A}} \over {\partial \phi}}} \right)^2} > 0.$$
(2.43)
Explicitly, one finds
$$\begin{array}{*{20}c} {({s_{ji}}\vert {s_{ji}}) = 2,\quad \quad \quad \quad \quad \quad} \\ {({\alpha _{ijk}}\vert {\alpha _{ijk}}) = 2,\quad \quad \quad \quad \quad \quad \quad} \\ {({e_{{i_1} \cdots {i_p}}}\vert {e_{{i_1} \cdots {i_p}}}) = {{p(d - p - 1)} \over {d - 1}} + {{{{\left({{\lambda ^{(p)}}} \right)}^2}} \over 4},} \\ {({m_{{i_1} \cdots {i_{p + 1}}}}\vert {m_{{i_1} \cdots {i_{p + 1}}}}) = {{p(d - p - 1)} \over {d - 1}} + {{{{\left({{\lambda ^{(p)}}} \right)}^2}} \over 4}.\quad \quad} \\ \end{array}$$
(2.44)
Because the potential walls are timelike, they have a non-empty intersection with the forward light cone in the space of the scale factors. When the null straight line representing the evolution of the scale factors hits one of the walls, it gets reflected according to the rule [43]
$${v^\mu}\quad \rightarrow \quad {v^\mu} - 2{{{v^\nu}{F_{A\nu}}} \over {({F_A}\vert {F_A})}}F_A^\mu ,$$
(2.45)
where v is the velocity vector (tangent to the straight line). This reflection preserves the time orientation since the hyperplanes are timelike and hence belong to the orthochronous Lorentz group O (k, 1) where k = d − 1 or d according to whether there is no or one dilaton. The conditions s ji = 0 define the “symmetry” or “centrifugal” walls, the conditions β ijk = 0 define the “curvature” or “gravitational” walls, the conditions \({\alpha _{ijk}} = 0\) define the “electric” walls, while the conditions \({m_{{i_1} \cdots {i_{p + 1}}}} = 0\) define the “magnetic” walls.

The motion is thus a succession of future-oriented null straight line segments interrupted by reflections against the walls, where the motion undergoes a reflection belonging to O(k, 1). Whether the collisions eventually stop or continue forever is better visualized by projecting the motion radially on the positive sheet of the unit hyperboloid, as was done first in the pioneering work of Chitre and Misner [31, 138] for pure gravity in four spacetime dimensions. We recall that the positive sheet of the unit hyperboloid ∑ (β i )2β i )2 + ϕ2 = −1, ∑β i > 0, provides a model of hyperbolic space (see, e.g., [146]).

The intersection of a timelike hyperplane with the unit hyperboloid defines a hyperplane in hyperbolic space. The region in hyperbolic space on the positive side of all hyperplanes is the allowed dynamical region and is called the “billiard table”. It is never compact in the cases relevant to gravity, but it may or may not have finite volume. The projection of the motion of the scale factors on the unit hyperboloid is the same as the motion of a billiard ball in a hyperbolic billiard: geodesic arcs in hyperbolic space within the billiard region, interrupted by collisions against the bounding walls where the motion undergoes a specular reflection.

When the volume of the billiard table is finite, the collisions with the potential walls never end (for generic initial data) and the motion is chaotic. When, on the other hand, the volume is infinite, generic initial data lead to a motion that ultimately freely runs away to infinity. This is non-chaotic. For more information, see [135, 170]. An interesting criterion for chaos (equivalent to finite volume of hyperbolic billiard region) has been given in [111] in terms of illuminations of spheres by point sources.

2.4.1 Comments

  1. 1.

    The task of determining the billiard region is greatly simplified by the observation that some walls are behind others and are thus not relevant. For instance, it is clear that if β2β1 > 0 and β3β2 > 0, then β3β1 > 0. Among the symmetry wall conditions, the only relevant ones are \({\beta ^{i + 1}} - {\beta ^i} > 0, \, i = 1,\,2,\, \cdots, \, d - 1\)β i > 0, i = 1. 2. …, d − 1. Similarly, a wall of any given type can be written as a positive combination of the walls of the same type with smallest values of the indices i of the β’s and the symmetry walls (e.g., the electric wall condition β2 > 0 for a 1-form with zero dilaton coupling can be written as β1 + (β2β1) > 0 and is thus a consequence of β1 > 0 and β2β1 > 0). Finally, one also verifies that in the presence of true p-forms (0 < p < d − 1), the gravitational walls are never relevant as they can be written as combinations of p-form walls with positive coefficients [49].

     
  2. 2.

    It is interesting to determine the spatially homogeneous models that reproduce asymptotically the correct billiard limit. It is clear that in order to do so, homogeneous cosmological models need only contain the relevant walls. It is not necessary that they yield all the walls. Which homogeneity groups are acceptable depends on the system at hand. We list here a few examples. For vacuum gravity in four spacetime dimensions, the appropriate homogeneous models are the so-called Bianchi VIII or IX models. For vacuum gravity in higher dimensions, the structure constants of the homogeneity group must fulfill the conditions of [60] and the metric must include off-diagonal components (see also [58]). In the presence of a single p-form and no dilaton (0 < p < d − 1), the simplest (Abelian) homogeneity group can be taken [44].

     

2.5 Rules for deriving the wall forms from the Lagrangian — Summary

We have recalled above that the generic behavior near a spacelike singularity of the system with action (2.1) can be described at each spatial point in terms of a billiard in hyperbolic space. The action for the billiard ball reads, in the gauge \(N = \sqrt g\),
$$S = \int d {x^0}\left[ {{G_{\mu \nu}}{{d{\beta ^\mu}} \over {d{x^0}}}{{d{\beta ^\nu}} \over {d{x^0}}} - V({\beta ^\mu})} \right],$$
(2.46)
where we recall that x0 in the BKL-limit (proper time T → 0+), and G μν is the metric in the space of the scale factors,
$${G_{\mu \nu}}\;d{\beta ^\mu}\;d{\beta ^\nu} = \sum\limits_{i = 1}^d d {\beta ^i}\;d{\beta ^i} - \left({\sum\limits_{i = 1}^d d {\beta ^i}} \right)\left({\sum\limits_{j = 1}^d d {\beta ^j}} \right) + d\phi \;d\phi$$
(2.47)
introduced in Equation (2.15) above. As stressed there, this metric is flat and of Lorentzian signature. Between two collisions, the motion is a free, geodesic motion. The collisions with the walls are controlled by the potential V(β μ ), which is a sum of sharp wall potentials. The walls are hyperplanes and can be inferred from the Lagrangian. They are as follows:
  1. 1.
    Gravity brings in the symmetry walls
    $${\beta ^{i + 1}} - {\beta ^i} = 0,$$
    (2.48)
    with i = 1, 2, …,d − 1, and the curvature wall
    $$2{\beta ^1} + {\beta ^2} + \cdots + {\beta ^{d - 2}} = 0.$$
    (2.49)
     
  2. 2.
    Each p-form brings in an electric wall
    $${\beta ^1} + \cdots + {\beta ^p} + {{{\lambda ^{(p)}}} \over 2}\phi = 0,$$
    (2.50)
    and a magnetic wall
    $${\beta ^1} + \cdots + {\beta ^{d - p - 1}} - {{{\lambda ^{(p)}}} \over 2}\phi = 0.$$
    (2.51)
     
We have written here only the (potentially) relevant walls. There are other walls present in the potential, but because these are behind the relevant walls, which are infinitely steep in the BKL-limit, they are irrelevant. They are relevant, however, when trying to exhibit the symmetry in a complete treatment where the BKL-limit is the zeroth order term in a gradient expansion yet to be understood [47].
The scalar product dual to the scalar product in the space of the scale factors is
$$(F\vert G) = \sum\limits_i {{F_i}} {G_i} - {1 \over {d - 1}}\left({\sum\limits_i {{F_i}}} \right)\left({\sum\limits_j {{G_j}}} \right) + {F_\phi}{G_\phi}$$
(2.52)
for two linear forms \(F = {F_i}{\beta ^i} + {F_\phi}\phi, \, G = {G_i}{\beta ^i} + {G_\phi}\phi\).

These recipes are all that we shall need for investigating the regularity properties of the billiards associated with the class of actions Equation (2.1).

2.6 More on the free motion: The Kasner solution

The free motion between two bounces is a straight line in the space of the scale factors. In terms of the original metric components, it takes the form of the Kasner solution with dilaton. Indeed, the free motion is given by
$${\beta ^\mu} = {q^\mu}{x^0} + \beta _0^\mu ,$$
where the “velocities” q μ are subject to
$$\sum\limits_i {{{({q^i})}^2}} - {\left({\sum\limits_i {{q^i}}} \right)^2} + q_\phi ^2 = 0,$$
since the motion is lightlike by the Hamiltonian constraint. The proper time \(dT = - \sqrt g d{x^0}\) is then T = B exp(−Kx0), with K = ∑ i q i and for some constant B (we assume, as before, that the singularity is at T = 0+). Redefining then
$${p^\mu} = {{{q^\mu}} \over {\sum\nolimits_i {{q^i}}}}$$
yields the celebrated Kasner solution
$$d{s^2} = - d{T^2} + \sum\limits_i {{T^{2{p^i}}}} {\left({d{x^i}} \right)^2},$$
(2.53)
$$\phi = - {p_\phi}\ln T + A,$$
(2.54)
subject to the constraints
$$\sum\limits_i {{p^i}} = 1,\qquad \sum\limits_i {{{({p^i})}^2}} + p_\phi ^2 = 1,$$
(2.55)
where A is a constant of integration and where the coordinates x i have been suitably rescaled (if necessary).

2.7 Chaos and billiard volume

With our rules for writing down the billiard region, one can determine in which case the volume of the billiard is finite and in which case it is infinite. The finite-volume, chaotic case is also called “mixmaster case”, a terminology introduced in four dimensions in [137].

The following results have been obtained:
  • Pure gravity in D ≤ 10 dimensions is chaotic, but ceases to be so for D ≥ 11 [63, 62].

  • The introduction of a dilaton removes chaos [15, 3]. The gravitational four-derivative action in four dimensions, based on R2, is dynamically equivalent to Einstein gravity coupled to a dilaton [160]. Hence, chaos is removed also for this case.

  • p-form gauge fields (0 < p < d − 1) without scalar fields lead to a finite-volume billiard [44].

  • When both p-forms and dilatons are included, the situation is more subtle as there is a competition between two opposing effects. One can show that if the dilaton couplings are in a “subcritical” open region that contains the origin — i.e., “not too big” — the billiard volume is infinite and the system is non chaotic. If the dilaton couplings are outside of that region, the billiard volume is finite and the system is chaotic [49].

2.8 A note on the constraints

We have focused in the above presentation on the dynamical equations of motion. The constraints were only briefly mentioned, with no discussion, except for the Hamiltonian constraint. This is legitimate because the constraints are first class and hence preserved by the Hamiltonian evolution. Thus, they need only be imposed at some “initial” time. Once this is done, one does not need to worry about them any more. Furthermore the momentum constraints and Gauss’ law constraints are differential equations relating the initial data at different spatial points. This means that they do not constrain the dynamical variables at a given point but involve also their gradients — contrary to the Hamiltonian constraint which becomes ultralocal. Consequently, at any given point, one can freely choose the initial data on the undifferentiated dynamical variables and then use these data as (part of) the appropriate boundary data necessary to integrate the constraints throughout space. This is why one can assert that all the walls described above are generically present even when the constraints are satisfied.

The situation is different in homogeneous cosmologies where the symmetry relates the values of the fields at all spatial points. The momentum and Gauss’ law constraints become then algebraic equations and might remove some relevant walls. But this feature (removal of walls by the momentum and Gauss’ law constraints) is specific to some homogeneous cosmologies and does not hold in the generic case where spatial gradients are non-zero.

A final comment: How the spatial diffeomorphism constraints and Gauss’ law fit in the conjectured infinite-dimensional symmetry is a point that is still poorly understood. See, however, [52] for recent progress in this direction.

2.9 On the validity of the BKL conjecture — A status report

Providing a complete rigorous justification of the above description of the behavior of the gravitational field in the vicinity of a spacelike singularity is a formidable task that has not been pushed to completion yet. The task is formidable because the Einstein equations form a complicated nonlinear system of partial differential equations. We shall assume throughout our review that the BKL description is correct, based on the original convincing arguments put forward by BKL themselves [16] and the subsequent fruitful investigations that have shed further important light on the validity of the conjecture. The billiard description will thus be taken for granted.

For completeness, we provide in this section a short guide to the work that has been accumulated since the late 1960’s to consolidate the BKL phenomenon.

As we have indicated, there are two aspects to the BKL conjecture:
  1. 1.

    The first part of the conjecture states that spatial points decouple as one goes to a spacelike singularity in the sense that the evolution can be described by a collection of systems of ordinary differential equations with respect to time, one such system at each spatial point. (“A spacelike singularity is local.”)

     
  2. 2.

    The second part of the conjecture states that the system of ordinary differential equations with respect to time describing the asymptotic dynamics at any given spatial point can be asymptotically replaced by the billiard equations. If the matter content is such that the billiard table has infinite volume, the asymptotic behavior at each point is given by a (generalized) Kasner solution (“Kasner-like spacelike singularities”). If, on the other hand, the matter content is such that the billiard table has finite volume, the asymptotic behavior at each point is a chaotic, infinite, oscillatory succession of Kasner epochs. (“Oscillatory, or mixmaster, spacelike singularities.”)

     
A third element of the original conjecture was that the matter could be neglected asymptotically. While generically true in four spacetime dimensions (the exception being a massless scalar field, equivalent to a fluid with the stiff equation of state p = ρ), this aspect of the conjecture does not remain valid in higher dimensions where the p-form fields might add relevant walls that could change the qualitative asymptotic behavior. We shall thus focus here only on Aspects 1 and 2.
  • In the Kasner-like case, the mathematical situation is easier to handle since the conjectured asymptotic behavior of the fields is then monotone and known in closed form. There exist theorems validating (generically) this conjectured asymptotic behavior, starting from the pioneering work of [3] (where the singularities with this behavior are called “quiescent”), which was extended later in [49] to cover more general matter contents. See also [18, 108] for related work.

  • The situation is much more complicated in the oscillatory case, where only partial results exist. However, even though as yet incomplete, the mathematical and numerical studies of the BKL analysis has provided overwhelming support for its validity. Most work has been done in four dimensions.

    The first attempts to demonstrate that spacelike singularities are local were done in the simpler context of solutions with isometries. It is only recently that general solutions without symmetries have been treated, but this has been found to be possible only numerically so far [87]. The literature on this subject is vast and we refer to [2, 87, 147] for points of entry into it. Let us note that an important element in the analysis has been a more precise reformulation of what is meant by “local”. This has been achieved in [163], where a precise definition involving a judicious choice of scale invariant variables has been proposed and given the illustrative name of “asymptotic silence” — the singularities being called “silent singularities” since propagation of information is asymptotically eliminated.

    If one accepts that generic spacelike singularities are silent, one can investigate the system of ordinary differential equations that arise in the local limit. In four dimensions, this system is the same as the system of ordinary differential equations describing the dynamics of spatially homogeneous cosmologies of Bianchi type IX. It has been effectively shown analytically in [151] that the Bianchi IX evolution equations can indeed be replaced, in the generic case, by the billiard equations (with only the dominant, sharp walls) that produce the mixmaster behavior. This validates the second element in the BKL conjecture in four dimensions.

The connection between the billiard variables and the scale invariant variables has been investigated recently in the interesting works [92, 162].

Finally, taking for granted the BKL conjecture, one might analyze the chaotic properties of the billiard map (when the volume is finite). Papers exploring this issue are [30, 32, 121, 132] (four dimensions) and [68] (five dimensions).

Let us finally mention the interesting recent paper [40], in which a more precise formulation of the BKL conjecture, aimed towards the chaotic case, is presented. In particular, the main result of this work is an extension of the Fuchsian techniques, employed, e.g., in [49], which are applicable also for systems exhibiting chaotic dynamics. Furthermore, [40] examines the geometric structure which is preserved close to the singularity, and it is shown that this structure has a mathematical description in terms of a so called “partially framed flag”.

3 Hyperbolic Coxeter Groups

In this section, we develop the theory of Coxeter groups with a particular emphasis on the hyperbolic case. The importance of Coxeter groups for the BKL analysis stems from the fact that in the case of the gravitational theories that have been studied most (pure gravity, supergravities), the group generated by the reflections in the billiard walls is a Coxeter group. This follows, in turn, from the regularity of the corresponding billiards, whose walls intersect at angles that are integer submultiples of π.

3.1 Preliminary example: The BKL billiard (vacuum D = 4 gravity)

To illustrate the regularity of the gravitational billiards and motivate the mathematical developments through an explicit example, we first compute in detail the billiard characterizing vacuum, D = 4 gravity. Since this corresponds to the case originally considered by BKL, we call it the “BKL billiard”. We show in detail that the billiard reflections in this case are governed by the “extended modular group” PGL(2. ℤ), which, as we shall see, is isomorphic to the hyperbolic Coxeter group \(A_1^{+ +}\).

3.1.1 Billiard reflections

There are three scale factors so that after radial projection on the unit hyperboloid, we get a billiard in two-dimensional hyperbolic space. The billiard region is defined by the following relevant wall inequalities,
$${\beta ^2} - {\beta ^1} > 0,\qquad {\beta ^3} - {\beta ^2} > 0$$
(3.1)
(symmetry walls) and
$$2{\beta ^1} > 0$$
(3.2)
(curvature wall). The remarkable properties of this region from our point of view are:
  • It is a triangle (i.e., a simplex in two dimensions) because even though we had to begin with 6 walls (3 symmetry walls and 3 curvature walls), only 3 of them are relevant.

  • The walls intersect at angles that are integer submultiples of π, i.e., of the form
    $${\pi \over n},$$
    (3.3)
    where n is an integer. The symmetry walls intersect indeed at sixty degrees (n = 3) since the scalar product of the corresponding linear forms (of norm squared equal to 2) is −1, while the gravitational wall makes angles of zero (n = ∞, scalar product = − 2) and ninety (n = 2, scalar product = 0) degrees with the symmetry walls.
These angles are captured in the matrix A = (A ij )i,j=1,2,3 of scalar products,
$${A_{ij}} = \left({{\alpha _i}\vert {\alpha _j}} \right),$$
(3.4)
which reads explicitly
$$A = \left({\begin{array}{*{20}c} 2 & {- 2} & 0 \\ {- 2} & 2 & {- 1} \\ 0 & {- 1} & 2 \\ \end{array}} \right).$$
(3.5)
Recall from the previous section that the scalar product of two linear forms F = F i β i and \(G = {G_i}{\beta ^i}\) is, in a three-dimensional scale factor space,
$$(F\vert G) = \sum\limits_i {{F_i}} {G_i} - {1 \over 2}\left({\sum\limits_i {{F_i}}} \right)\left({\sum\limits_i {{G_i}}} \right),$$
(3.6)
where we have taken α1(β) = 2β1, α2(β) = β2β1 and α3(β) = β3β2. The corresponding billiard region is drawn in Figure 1.
Figure 1

The BKL billiard of pure four-dimensional gravity. The figure represents the billiard region projected onto the hyperbolic plane. The particle geodesic is confined to the fundamental region enclosed by the three walls α1(β) = 2β1 = 0, α2 (β) = β2β1 = 0 and β3 (β) = β3β2 = 0, as indicated by the numbering in the figure. The two symmetry walls α2(β) = 0 and α3(β) = 0 intersect at an angle of π/3, while the gravity wall α1 (β) = 0 intersects, respectively, at angles 0 and π/2 with the symmetry walls α2(β) = 0 and α3(β) = 0. The particle has no direction of escape so the dynamics is chaotic.

Because the angles between the reflecting planes are integer submultiples of π, the reflections in the walls bounding the billiard region6,
$${s_i}(\gamma) = \gamma - 2{{(\gamma \vert {\alpha _i})} \over {({\alpha _i}\vert {\alpha _i})}}{\alpha _i} = \gamma - (\gamma \vert {\alpha _i}){\alpha _i},$$
(3.7)
obey the following relations,
$${s_1}{s_3} = {s_3}{s_1}\quad \leftrightarrow \quad {({s_1}{s_3})^2} = 1,\qquad {({s_2}{s_3})^3} = 1.$$
(3.8)
The product s1s3 is a rotation by 2π/2 = π and hence squares to one; the product s2s3 is a rotation by 2π/3 and hence its cube is equal to one. There is no power of the product s1s2 that is equal to one, something that one conventionally writes as
$${({s_1}{s_3})^\infty} = 1.$$
(3.9)

The group generated by the reflections s1, s2 and s3 is denoted \(A_1^{+ +}\), for reasons that will become clear in the following, and coincides with the arithmetic group PGL(2, ℤ), as we will now show (see also [75, 116, 107]).

3.1.2 On the group PGL(2, ℤ)

The group PGL(2. ℤ) is defined as the group of 2 × 2 matrices C with integer entries and determinant equal to ±1, with the identification of C and −C,
$$PGL(2,{\mathbb Z}) = {{GL(2,{\mathbb Z})} \over {{{\mathbb Z}_2}}}.$$
(3.10)
Note that although elements of the real general linear group GL(2, ℝ) have (non-vanishing) unrestricted determinants, the discrete subgroup GL(2. ℝ) ⊂ GL(2. ℝ) only allows for det C = ±1 in order for the inverse C−1 to also be an element of GL(2. ℤ).
There are two interesting realisations of PGL(2. ℤ) in terms of transformations in two dimensions:
  • One can view PGL(2. ℤ) as the group of fractional transformations of the complex plane
    $$C:z \rightarrow z\prime = {{az + b} \over {cz + d}},\qquad a,b,c,d \in {\mathbb Z},$$
    (3.11)
    with
    $$ad - cd = \pm 1.$$
    (3.12)
    Note that one gets the same transformation if C is replaced by −C, as one should. It is an easy exercise to verify that the action of PGL(2. ℤ) when defined in this way maps the complex upper half-plane,
    $${\rm{{\mathbb H}}} = \{z \in {\rm{{\mathbb C}}}\,\vert \,\Im z > 0\} ,$$
    (3.13)
    onto itself whenever the determinant adbc of C is equal to +1. This is not the case, however, when det C = −1.
  • For this reason, it is convenient to consider alternatively the following action of PGL(2. ℤ),
    $$\begin{array}{*{20}c} {z \rightarrow z\prime = {{az + b} \over {cz + d}},\quad \,\;\;\;\;\;\;{\rm{if}}\;ad - cb = 1,} \\ {{\rm{or}}} \\ {z \rightarrow z\prime = {{a\bar z + b} \over {c\bar z + d}},\quad \quad \,{\rm{if}}\;ad - cb = - 1,} \\\end{array}$$
    (3.14)
    (a. b. c. d ∈ ℤ), which does map the complex upper-half plane onto itself, i.e., which is such that \(\mathfrak I {z\prime} > 0\) whenever \(\mathfrak I z > 0\).

    The transformation (3.14) is the composition of the identity with the transformation (3.11) when det C =1, and of the complex conjugation transformation, \(f:z \rightarrow \bar z\) with the transformation (3.11) when det C = −1. Because the coefficients a, b, c, and d are real, f commutes with C and furthermore the map (3.11)(3.14) is a group isomorphism, so that we can indeed either view the group PGL(2, →) as the group of fractional transformations (3.11), or as the group of transformations (3.14).

An important subgroup of the group PGL(2, ℤ) is the group PSL(2, ℤ) for which adcb = 1, also called the “modular group”. The translation T: zz +1 and the inversion S: z → −1/z are examples of modular transformations,
$$\begin{array}{*{20}c} {T = \left({\begin{array}{*{20}c} 1 & 1 \\ 0 & 1 \\ \end{array}} \right),} & {S = \left({\begin{array}{*{20}c} 0 & {- 1} \\ 1 & 0 \\ \end{array}} \right).} \\ \end{array}$$
(3.15)
It is a classical result that any modular transformation can be written as the product
$${T^{{m_1}}}S{T^{{m_2}}}S \cdots S{T^{{m_k}}},$$
(3.16)
but the representation is not unique [4].
Let s1, s2 and s3 be the PGL(2, ℤ)-transformations
$$\begin{array}{*{20}c} {{s_1}:z} & \rightarrow & {- \bar z,} \\ {{s_2}:z} & \rightarrow & {1 - \bar z,} \\ {{s_3}:z} & \rightarrow & {{1 \over z},} \\ \end{array}$$
(3.17)
to which there correspond the matrices
$${s_1} = \left({\begin{array}{*{20}c} 1 & 0 \\ 0 & {- 1} \\ \end{array}} \right),\quad \quad {s_2} = \left({\begin{array}{*{20}c} 1 & {- 1} \\ 0 & {- 1} \\ \end{array}} \right),\quad \quad {s_3} = \left({\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array}} \right)$$
(3.18)
The s i ’s are reflections in the straight lines x = 0, x =1/2 and the unit circle |z| = 1, respectively. These are in fact just the transformations of hyperbolic space s1, s2 and s3 described in Section 3.1.1, since the reflection lines intersect at 0, 90 and 60 degrees, respectively.
One easily verifies that T = s2s1 and that S = s1s3 = s3s1. Since any transformation of PGL(2, ℤ) not in PSL(2, ℤ) can be written as a transformation of PSL(2, ℤ) times, say, s1 and since any transformation of PSL(2, ℤ) can be written as a product of S’s and T’s, it follows that the group generated by the 3 reflections s1, s2 and s3 coincides with PGL(2, ℤ), as announced above. (Strictly speaking, PGL(2, ℤ) could be a quotient of that group by some invariant subgroup, but one may verify that the kernel of the homomorphism is trivial (see Section 3.2.5 below).) The fundamental domains for PGL(2, ℤ) and PSL(2, ℤ) are drawn in Figure 2. The equivalence between PGL(2, ℤ) and the Coxeter group \(A_1^{+ +}\) has been discussed previously in [75, 116, 107].
Figure 2

The figure on the left hand side displays the action of the modular group PSL(2, ℤ) on the complex upper half plane \({\mathbb H}{\rm{= \{z}} \in {\mathbb C}{\vert}{\mathfrak I}{\rm{z > 0\}}}\). The two generators of PSL(2, ℤ) are S and T, acting as follows on the coordinate z ∈ : S(z) = − 1/z; T (z) = z + 1, i.e., as an inversion and a translation, respectively. The shaded area indicates the fundamental domain \({{\mathcal D}_{PSL(2,{\mathbb Z})}} = \{z \in {\mathbb H}\vert - 1/2 \leq {\mathfrak R}z \leq 1/2;\,\vert z \vert \geq 1\}\) for the action of PSL(2, ℤ) on ℍ. The figure on the right hand side displays the action of the “extended modular group” PGL (2, ℤ) on ℍ. The generators of PGL (2, ℤ) are obtained by augmenting the generators of PSL(2, ℤ) with the generator s1, acting as \({s_1}(z) = - \bar z\) on ℍ. The additional two generators of PGL(2, ℤ) then become: s2 = s1 oT; s3 = s1 oS, and their actions on ℍ are \({s_2}(z) = 1 - \bar z;\,{s_3}(z) = 1/\bar z\) The new generator s1 corresponds to a reflection in the line \({\mathfrak R}z{\rm{= 0}}\), the generator s2 is in turn a reflection in the line \({\mathfrak R}z = 1/2\), while the generator s3 is a reflection in the unit circle |z| = 1. The fundamental domain of PGL(2, ℍ) is \({{\mathcal D}_{PGL(2,{\mathbb Z})}} = \{z \in {\mathbb H} \vert 0 \leq {\mathfrak R}z \leq 1/2;\,\vert z \vert \geq 1\}\), corresponding to half the fundamental domain of PSL(2, ℤ). The “walls” \({\mathfrak R}z = 0,\,{\mathfrak R}z = 1/2\) and |z| = 1 correspond, respectively, to the gravity wall α1(β) = 0, the symmetry wall α2(β) = 0 and the symmetry wall α3(β) = 0 of Figure 1.

3.2 Coxeter groups — The general theory

We have just shown that the billiard group in the case of pure gravity in four spacetime dimensions is the group PGL(2, ℤ). This group is generated by reflections and is a particular example of a Coxeter group. Furthermore, as we shall explain below, this Coxeter group turns out to be the Weyl group of the (hyperbolic) Kac-Moody algebra \(A_1^{+ +}\). Our first encounter with Lorentzian Kac-Moody algebras in more general gravitational theories will also be through their Weyl groups, which are, exactly as in the four-dimensional case just described, particular instances of (non-Euclidean) Coxeter groups, and which arise as the groups of billiard reflections.

For this reason, we start by developing here some aspects of the theory of Coxeter groups. An excellent reference on the subject is [107], to which we refer for more details and information. We consider Kac-Moody algebras in Section 4.

3.2.1 Examples

Coxeter groups generalize the familiar notion of reflection groups in Euclidean space. Before we present the basic definition, let us briefly discuss some more illuminating examples.

3.2.1.1 The dihedral group I2(3) = A2

Consider the dihedral group I2(3) of order 6 of symmetries of the equilateral triangle in the Euclidean plane.

This group contains the identity, three reflections s1, s2 and s3 about the three medians, the rotation R1 of 2π/3 about the origin and the rotation R2 of 4π/3 about the origin (see Figure 3),
$${I_{\rm{2}}}{\rm{(3) \,=\, \{1,}}{{\rm{s}}_{\rm{1}}}{\rm{,}}{{\rm{s}}_{\rm{2}}}{\rm{,}}{{\rm{s}}_{\rm{3}}}{\rm{,}}{R_{\rm{1}}}{\rm{,}}{R_{\rm{2}}}{\rm{\}}}.$$
(3.19)
The reflections act as follows7,
$${s_i}(\gamma) = \gamma - 2{{(\gamma \vert {\alpha _i})} \over {({\alpha _i}\vert {\alpha _i})}}{\alpha _i},$$
(3.20)
where (|) is here the Euclidean scalar product and where α i is a vector orthogonal to the hyper-plane (here, line) of reflection.
Figure 3

The equilateral triangle with its 3 axes of symmetries. The reflections s1 and s2 generate the entire symmetry group. We have pictured the vectors α1 and α2 orthogonal to the axes of reflection and chosen to make an obtuse angle. The shaded region {w|(w|α1) ≥ 0} ∩ {w|(w|α2) ≥ 0} is a fundamental domain for the action of the group on the triangle. Note that the fundamental domain for the action of the group on the entire Euclidean plane extends indefinitely beyond the triangle but is, of course, still bounded by the two walls orthogonal to α1 and α2.

Now, all elements of the dihedral group I2(3) can be written as products of the two reflections s1 and s2:
$$1 = s_1^0,\quad \quad {s_1} = {s_1},\quad \quad {s_2} = {s_2},\quad \quad {R_1} = {s_1}{s_2},\quad \quad {R_2} = {s_2}{s_1}\quad \quad {s_3} = {s_1}{s_2}{s_1}.$$
(3.21)
Hence, the dihedral group I2(3) is generated by s1 and s2. The writing Equation (3.21) is not unique because s1 and s2 are subject to the following relations,
$$S_1^2 = 1,\quad \quad s_2^2 = 1,\quad \quad {({s_1}{s_2})^3} = 1.$$
(3.22)
The first two relations merely follow from the fact that s1 and s2 are reflections, while the third relation is a consequence of the property that the product s1s2 is a rotation by an angle of 2π/3. This follows, in turn, from the fact that the hyperplanes (lines) of reflection make an angle of π/3. There is no other relation between the generators s1 and s2 because any product of them can be reduced, using the relations Equation (3.22), to one of the 6 elements in Equation (3.21), and these are independent.

The dihedral group I2(3) is also denoted A2 because it is the Weyl group of the simple Lie algebra A2 (see Section 4). It is isomorphic to the permutation group S3 of three objects.

3.2.1.2 The infinite dihedral group \({I_2}(\infty) \equiv A_1^ +\)
Consider now the group of isometries of the Euclidean line containing the symmetries about the points with integer or half-integer values of x (x is a coordinate along the line) as well as the translations by an integer. This is clearly an infinite group. It is generated by the two reflections s1 about the origin and s2 about the point with coordinate 1/2,
$${S_1}(x) = - x,\quad \quad {s_2}(x) = - (x - 1).$$
(3.23)
The product s2s1 is a translation by +1 while the product s1s2 is a translation by −1, so no power of s1s2 or s2s1 gives the identity. All the powers (s2s1) k and (s1s2) j are distinct (translations by +k and −j, respectively). The only relations between the generators are
$$s_1^2 = 1 = s_2^2$$
(3.24)
This infinite dihedral group I2() is also denoted by \(A_1^ +\) because it is the Weyl group of the affine Kac-Moody algebra \(A_1^ +\).

3.2.2 Definition

A Coxeter group \({\mathfrak C}\) is a group generated by a finite number of elements s i (i = 1. …,n) subject to relations that take the form
$$s_i^2 = 1$$
(3.25)
and
$${({s_i}{s_j})^{{m_{ij}}}} = 1,$$
(3.26)
where the integers m ij associated with the pairs (i, j) fulfill
$$\begin{array}{*{20}c} {{m_{ij}} = {m_{ji}},\quad \quad \quad \;} \\ {{m_{ij}} \geq 2\quad \quad (i \neq j){.}} \\ \end{array}$$
(3.27)
Note that Equation (3.25) is a particular case of Equation (3.26) with m ii = 1. If there is no power of s i s j that gives the identity, as in our second example, we set, by convention, m ij = . The generators s i are called “reflections” because of Equation (3.25), even though we have not developed yet a geometric realisation of the group. This will be done in Section 3.2.4 below.

The number n of generators is called the rank of the Coxeter group. The Coxeter group is completely specified by the integers m ij . It is useful to draw the set {m ij } pictorially in a diagram Γ, called a Coxeter graph. With each reflection s i , one associates a node. Thus there are n nodes in the diagram. If m ij > 2, one draws a line between the node i and the node j and writes m ij over the line, except if m ij is equal to 3, in which case one writes nothing. The default value is thus “3”. When there is no line between i and j (ij), the exponent m ij is equal to 2. We have drawn the Coxeter graphs for the Coxeter groups I2(3), I2(m) and for the Coxeter group H3 of symmetries of the icosahedron.

Note that if m ij = 2, the generators s i and s j commute, s i s j = s i s j . Thus, a Coxeter group \({\mathfrak C}\) is the direct product of the Coxeter subgroups associated with the connected components of its Coxeter graph. For that reason, we can restrict the analysis to Coxeter groups associated with connected (also called irreducible) Coxeter graphs.

The Coxeter group may be finite or infinite as the previous examples show.

3.2.2.1 Another example: \(C_2^ +\)

It should be stressed that the Coxeter group can be infinite even if none of the Coxeter exponent is infinite. Consider for instance the group of isometries of the Euclidean plane generated by reflections in the following three straight lines: (i) the x-axis (s1), (ii) the straight line joining the points (1,0) and (0,1) (s2), and (iii) the y-axis (s3). The Coxeter exponents are finite and equal to 4 (m12 = m21 = m23 = m32 = 4) and 2 (m13 = m31 = 2). The Coxeter graph is given in Figure 7. The Coxeter group is the symmetry group of the regular paving of the plane by squares and contains translations. Indeed, the product s2s1s2 is a reflection in the line parallel to the y-axis going through (1, 0) and thus the product t = s2 s1s2s3 is a translation by +2 in the x-direction. All powers of t are distinct; the group is infinite. This Coxeter group is of affine type and is called \(C_2^ +\) (which coincides with \(B_2^ +\))

3.2.2.2 The isomorphism problem
The Coxeter presentation of a given Coxeter group may not be unique. Consider for instance the group I2(6) of order 12 of symmetries of the regular hexagon, generated by two reflections s1 and s2 with
$$s_1^2 = s_2^2 = 1,\quad \quad {({s_1}{s_2})^6} = 1.$$
This group is isomorphic with the rank 3 (reducible) Coxeter group I2 (3) × ℤ2, with presentation
$$r_1^2 = r_2^2 = r_3^2 = 1,\quad \quad {({r_1}{r_2})^3} = 1,\quad \quad {({r_1}{r_3})^2} = 1,\quad \quad {({r_2}{r_3})^2} = 1,$$
the isomorphism being given by f(r1) = s1, f(r2) = s1s2s1s2s1, f(r3) = (s1s2)3. The question of determining all such isomorphisms between Coxeter groups is known as the “isomorphism problem of Coxeter groups”. This is a difficult problem whose general solution is not yet known [10].

3.2.3 The length function

An important concept in the theory of Coxeter groups is that of the length of an element. The length of \(w \in {\mathfrak C}\) is by definition the number of generators that appear in a minimal representation of w as a product of generators. Thus, if \(w = {s_{{i_1}}}\,{s_{{i_2}}}\, \cdots {s_{{i_l}}}\) and if there is no way to write w as a product of less than l generators, one says that w has length l.

For instance, for the dihedral group I2(3), the identity has length zero, the generators s1 and s2 have length one, the two non-trivial rotations have length two, and the third reflection s3 has length three. Note that the rotations have representations involving two and four (and even a higher number of) generators since for instance s1s2 = s2s1s2s1, but the length is associated with the representations involving as few generators as possible. There might be more than one such representation as it occurs for s3 = s1s2s1 = s2s1 s2. Both involve three generators and define the length of s3 to be three.

Let w be an element of length l. The length of ws i (where s i is one of the generators) differs from the length of w by an odd (positive or negative) integer since the relations among the generators always involve an even number of reflections. In fact, l(ws i ) is equal to l + 1 or l − 1 since l(ws i ) ≤ l(w) + 1 and l(w = ws i s j ) ≤ l(ws i ) + 1. Thus, in ws i , there can be at most one simplification (i.e., at most two elements that can be removed using the relations).

3.2.4 Geometric realization

We now construct a geometric realisation for any given Coxeter group. This enables one to view the Coxeter group as a group of linear transformations acting in a vector space of dimension n, equipped with a scalar product preserved by the group.

To each generator s i , associate a vector α i of a basis {α1, …, α n } of an n-dimensional vector space V. Introduce a scalar product defined as follows,
$$B({\alpha _i},{\alpha _j}) = - \cos ({\pi \over {{m_{ij}}}}),$$
(3.28)
on the basis vectors and extend it to V by linearity. Note that for i = j, m ii = 1 implies B(α i ,α i ) = 1 for all i. In the case of the dihedral group A2, this scalar product is just the Euclidean scalar product in the two-dimensional plane where the equilateral triangle lies, as can be seen by taking the two vectors α1 and α2 respectively orthogonal to the first and second lines of reflection in Figure 3 and oriented as indicated. But in general, the scalar product (3.28) might not be of Euclidean signature and might even be degenerate. This is the case for the infinite dihedral group I2(), for which the matrix B reads
$$B = \left({\begin{array}{*{20}c} 1 & {- 1} \\ {- 1} & 1 \\ \end{array}} \right)$$
(3.29)
and has zero determinant. We shall occasionally use matrix notations for the scalar product, B(α,γ) = α T .
However, the basis vectors are always all spacelike since they have norm squared equal to 1. For each i, the vector space V splits then as a direct sum
$$V = {\rm{{\mathbb R}}}{\alpha _i} \oplus {H_i},$$
(3.30)
where H i is the hyperplane orthogonal to α i (δH i iff B(γ, α i ) =0). One defines the geometric reflection σ i as
$${\sigma _i}(\gamma) = {\rm{}}\gamma - 2B(\gamma ,{\alpha _i}){\alpha _i}.$$
(3.31)
It is clear that σ i fixes H i pointwise and reverses α i . It is also clear that \(\sigma _i^2 = 1\) and that σ i preserves B,
$$B\left({{\sigma _i}(\gamma),{\sigma _i}(\gamma \prime)} \right) = B(\gamma ,\gamma \prime)$$
(3.32)
Note that in the particular case of A2, we recover in this way the reflections s1 and s2.
We now verify that the σ i ’s also fulfill the relations \({({\sigma _i}{\sigma _j})^{{m_{ij}}}} = 1\). To that end we consider the plane Π spanned by α i and α j . This plane is left invariant under σ i and σ j . Two possibilities may occur:
  1. 1.

    The induced scalar product on Π is nondegenerate and in fact positive definite, or

     
  2. 2.

    the induced scalar product is positive semi-definite, i.e., there is a null direction orthogonal to any other direction.

     
The second case occurs only when m ij = . The null direction is given by γ = α i + α j .
  • In Case 1, V splits as \({\sigma _i}{\sigma _j}{)^{{m_{ij}}}}\) is clearly the identity on Π since both σ i and σ j leave Π pointwise invariant. One needs only to investigate \({\sigma _i}{\sigma _j}{)^{{m_{ij}}}}\) on Π, where the metric is positive definite. To that end we note that the reflections σ i and σ j are, on Π, standard Euclidean reflections in the lines orthogonal to α i and α j , respectively. These lines make an angle of π/m ij and hence the product σ i σ j is a rotation by an angle of 2π/m ij . It follows that \({({\sigma _i}{\sigma _j})^{{m_{ij}}}} = 1\) also on Π.

  • In Case 2, \({{m_{ij}}}\) is infinite and we must show that no power of the product σ i σ j gives the identity. This is done by exhibiting a vector γ for which (σ i σ j ) k (γ) ≠ γ for all integers k different from zero. Take for instance α i . Since one has (σ i σ j )(α i ) = α i + 2λ and (σ i σ j )(λ) = λ, it follows that (σ i σ j ) k (α i ) = α i + 2α i unless k = 0.

As the defining relations are preserved, we can conclude that the map f from the Coxeter group generated by the s i ’s to the geometric group generated by the σ i ’s defined on the generators by f (s i ) = σ i is a group homomorphism. We will show below that its kernel is the identity so that it is in fact an isomorphism.

Finally, we note that if the Coxeter graph is irreducible, as we assume, then the matrix B ij is indecomposable. A matrix A ij is called decomposable if after reordering of its indices, it decomposes as a non-trivial direct sum, i.e., if one can slit the indices i, j in two sets J and Λ such that A ij = 0 whenever iJ, j ∈ Λ or i ∈ Λ, jJ. The indecomposability of B follows from the fact that if it were decomposable, the corresponding Coxeter graph would be disconnected as no line would join a point in the set Λ to a point in the set J.

3.2.5 Positive and negative roots

A root is any vector in the space V of the geometric realisation that can be obtained from one of the basis vectors σ i by acting with an element w of the Coxeter group (more precisely, with its image f(w) under the above homomorphism, but we shall drop “f” for notational simplicity). Any root α can be expanded in terms of the α i ’s,
$$\alpha = \sum\limits_i {{c_i}} {\alpha _i}.$$
(3.33)
If the coefficients c i are all non-negative, we say that the root a is positive and we write α > 0. If the coefficients c i are all non-positive, we say that the root a is negative and we write α < 0. Note that we use strict inequalities here because if c i = 0 for all i, then a is not a root. In particular, the α i ’s themselves are positive roots, called also “simple” roots. (Note that the simple roots considered here differ by normalization factors from the simple roots of Kac-Moody algebras, as we shall discuss below.) We claim that roots are either positive or negative (there is no root with some c i ’s in Equation (3.33) > 0 and some other c i ’s < 0). The claim follows from the fact that the image of a simple root by an arbitrary element w of the Coxeter group is necessarily either positive or negative.

This, in turn, is the result of the following theorem, which provides a useful criterion to tell whether the length l(ws i ) of ws i is equal to l(w) + 1 or l(w) − 1.

Theorem: l(ws i ) = l(w) + 1 if and only if w(α i ) > 0.

The proof is given in [107], page 111.

It easily follows from this theorem that l(ws i ) = l(w) − 1 if and only if w(α i ) < 0. Indeed, l(ws i ) = l(w) − 1 is equivalent to l(w) = l(ws i ) + 1, i.e., l((ws i )s i ) = l(ws i ) + 1 and thus, by the theorem, ws i (α i ) > 0. But since s i (α i ) = −α i , this is equivalent to w(α i ) < 0.

We have seen in Section 3.2.3 that there are only two possibilities for the length l(ws i ). It is either equal to l(w) + 1 or to l(w) − 1. From the theorem just seen, the root w(α i ) is positive in the first case and negative in the second. Since any root is the Coxeter image of one of the simple roots α i , i.e., can be written as w(α i ) for some w and α i , we can conclude that the roots are either positive or negative; there is no alternative.

The theorem can be used to provide a geometric interpretation of the length function. One can show [107] that l(w) is equal to the number of positive roots sent by w to negative roots. In particular, the fundamental reflection s associated with the simple root α s maps α s to its negative and permutes the remaining positive roots.

Note that the theorem implies also that the kernel of the homomorphism that appears in the geometric realisation of the Coxeter group is trivial. Indeed, assume f(w) = 1 where w is an element of the Coxeter group that is not the identity. It is clear that there exists one group generator s i such that l(ws i ) = l(w) − 1. Take for instance the last generator occurring in a reduced expression of w. For this generator, one has w(α i ) < 0, which is in contradiction with the assumption f(w) = 1.

Because f is an isomorphism, we shall from now on identify the Coxeter group with its geometric realisation and make no distinction between s i and σ i .

3.2.6 Fundamental domain

In order to describe the action of the Coxeter group, it is useful to introduce the concept of fundamental domain. Consider first the case of the symmetry group A2 of the equilateral triangle. The shaded region \(\mathcal F\) in Figure 4 contains the vectors γ such that B(α1, γ) ≥ 0 and B(α2, γ) ≤ 0. It has the following important property: Any orbit of the group A2 intersects \({\mathcal F}\) once and only once. It is called for this reason a “fundamental domain”. We shall extend this concept to all Coxeter groups. However, when the scalar product B is not positive definite, there are inequivalent types of vectors and the concept of fundamental domain can be generalized a priori in different ways, depending on which region one wants to cover. (The entire space? Only the timelike vectors? Another region?) The useful generalization turns out not to lead to a fundamental domain of the action of the Coxeter group on the entire vector space V, but rather to a fundamental domain of the action of the Coxeter group on the so-called Tits cone \({\mathcal X}\), which is such that the inequalities B(α i , γ) ≥ 0 continue to play the central role.
Figure 4

The Coxeter graph of the symmetry group I2(3) = A2 of the equilateral triangle.

Figure 5

The Coxeter graph of the dihedral group I2(m).

Figure 6

The Coxeter graph of the symmetry group H3 of the regular icosahedron.

Figure 7

The Coxeter graph of the affine Coxeter group \(C_2^ +\) corresponding to the group of isometries of the Euclidean plane.

We assume that the scalar product is nondegenerate. Define for each simple root α i the open half-space
$${A_i} = \{\gamma \in V\,\vert \,B({\alpha _i},\gamma) > 0\} .$$
(3.34)
We define \({\mathcal E}\) to be the intersection of all A i ,
$${\mathcal E} = \bigcap\limits_i {{A_i}} .$$
(3.35)
This is a convex open cone, which is non-empty because the metric is nondegenerate. Indeed, as B is nondegenerate, one can, by a change of basis, assume for simplicity that the bounding hyperplanes B(α i , γ) = 0 are the coordinate hyperplanes x i = 0. \({\mathcal E}\) is then the region x i > 0 (with appropriate orientation of the coordinates) and \({\mathcal F}\) is x i ≥ 0. The closure
$$\begin{array}{*{20}c} {{\mathcal F}\;\; = \;\;\overline {\mathcal E} = \bigcap\nolimits_i {{{\bar A}_i}} ,\quad \quad \quad \quad} \\ {{{\bar A}_i}\;\; = \;\;\{\gamma \in V\vert B({\alpha _i},\gamma) \geq 0\}} \\ \end{array}$$
(3.36)
is then a closed convex cone8.
We next consider the union of the images of \(\mathcal F\) under the Coxeter group,
$${\mathcal X} = \bigcup\limits_{w \in {\mathfrak C}} w ({\mathcal F}).$$
(3.37)
One can show [107] that this is also a convex cone, called the Tits cone. Furthermore, \({\mathcal F}\) is a fundamental domain for the action of the Coxeter group on the Tits cone; the orbit of any point in \({\mathcal X}\) intersects \({\mathcal F}\) once and only once [107]. The Tits cone does not coincide in general with the full space V and is discussed below in particular cases.

3.3 Finite Coxeter groups

An important class of Coxeter groups are the finite ones, like I2(3) above. One can show that a Coxeter group is finite if and only if the scalar product defined by Equation (3.28) on V is Euclidean [107]. Finite Coxeter groups coincide with finite reflection groups in Euclidean space (through hyperplanes that all contain the origin) and are discrete subgroups of O(n). The classification of finite Coxeter groups is known and is given in Table 1 for completeness. For finite Coxeter groups, one has the important result that the Tits cone coincides with the entire space V [107].

3.4 Affine Coxeter groups

Affine Coxeter groups are by definition such that the bilinear form B is positive semi-definite but not positive definite. The radical V (defined as the subspace of vectors x for which B(x,y) = x T By = 0 for all y) is then one-dimensional (in the irreducible case). Indeed, since B is positive semi-definite, its radical coincides with the set N of vectors such that λT = 0 as can easily be seen by going to a basis in which B is diagonal (the eigenvalues of B are non-negative). Furthermore, N is at least one-dimensional since B is not positive definite (one of the eigenvalues is zero). Let μ be a vector in = N. Let ν be the vector whose components are the absolute values of those of μ, ν i = |μ i |Because B ij ≤ 0 for ij (see definition of B in Equation (3.28)), one has
$$0 \leq {\nu ^T}B\nu \leq {\mu ^T}B\mu = 0$$
and thus the vector ν belongs also to V. All the components of ν are strictly positive, ν i > 0. Indeed, let J be the set of indices for which ν j > 0 and I the set of indices for which ν i = 0. From j B kj ν j = 0 (νV) one gets, by taking k in I, that B ij = 0 for all iI, jJ, contrary to the assumption that the Coxeter system is irreducible (B is indecomposable). Hence, none of the components of any zero eigenvector μ can be zero. If V were more than one-dimensional, one could easily construct a zero eigenvector of B with at least one component equal to zero. Hence, the eigenspace V of zero eigenvectors is one-dimensional.
Affine Coxeter groups can be identified with the groups generated by affine reflections in Euclidean space (i.e., reflections through hyperplanes that may not contain the origin, so that the group contains translations) and have also been completely classified [107]. The translation subgroup of an affine Coxeter group \({\mathfrak C}\) is an invariant subgroup and the quotient \({{\mathfrak C}_0}\) is finite; the affine Coxeter group \({\mathfrak C}\) is equal to the semi-direct product of its translation subgroup by \({{\mathfrak C}_0}\)0. We list all the affine Coxeter groups in Table 2.
Table 2

Affine Coxeter groups.

Name

Coxeter graph

\(A_1^ +\)

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\(A_n^ + \,(n > 1)\)

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\(B_n^ + \,(n > 2)\)

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\(C_n^ +\)

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\(D_n^ +\)

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\(G_2^ +\)

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\(F_4^ +\)

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\(E_6^ +\)

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\(E_7^ +\)

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\(E_8^ +\)

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3.5 Lorentzian and hyperbolic Coxeter groups

Coxeter groups that are neither of finite nor of affine type are said to be of indefinite type. An important property of Coxeter groups of indefinite type is the following. There exists a positive vector (c i ) such that j B ij c j is negative [116]. A vector is said to be positive (respectively, negative) if all its components are strictly positive (respectively, strictly negative). This is denoted c i > 0 (respectively, c i < 0). Note that a vector may be neither positive nor negative, if some of its components are positive while some others are negative. Note also that these concepts refer to a specific basis. This property is demonstrated in Appendix A.

We assume, as already stated, that the scalar product B is nondegenerate. Let {ω i } be the basis dual to the basis {α i } in the scalar product B,
$$B({\alpha _i},{\omega _j}) = {\delta _{ij}}.$$
(3.38)
The ω i ’s are called “fundamental weights”. (The fundamental weights are really defined by Equation (3.38) up to normalization, as we will see in Section 3.6 on crystallographic Coxeter groups. They thus differ from the solutions of Equation (3.38) only by a positive multiplicative factor, irrelevant for the present discussion.)

Consider the vector υ = i c i α i , where the vector c i is such that c i > 0 and j B ij c j < 0. This vector exists since we assume the Coxeter group to be of indefinite type. Let Σ be the hyperplane orthogonal to υ. Because c i > 0, the vectors ω i ’s all lie on the positive side of Σ, B(υ, ω i ) = c i > 0. By contrast, the vectors α i ’s all lie on the negative side of Σ since B(α i , υ) = j B ij c j < 0. Furthermore, υ has negative norm squared, B(υ, υ) = j c j ( j B ij c j ) < 0. Thus, in the case of Coxeter groups of indefinite type (with a nondegenerate metric), one can choose a hyperplane such that the positive roots lie on one side of it and the fundamental weights on the other side. The converse is true for Coxeter group of finite type: In that case, there exists c i > 0 such that j B ij c j is positive, implying that the positive roots and the fundamental weights are on the same side of the hyperplane Σ.

We now consider a particular subclass of Coxeter groups of indefinite type, called Lorentzian Coxeter groups. These are Coxeter groups such that the scalar product B is of Lorentzian signature (n − 1,1). They are discrete subgroups of the orthochronous Lorentz group O+(n − 1, 1) preserving the time orientation. Since the α i are spacelike, the reflection hyperplanes are timelike and thus the generating reflections s i preserve the time orientation. The hyperplane Σ from the previous paragraph is spacelike. In this section, we shall adopt Lorentzian coordinates so that Σ has equation x0 = 0 and we shall choose the time orientation so that the positive roots have a negative time component. The fundamental weights have then a positive time component. This choice is purely conventional and is made here for convenience. Depending on the circumstances, the other time orientation might be more useful and will sometimes be adopted later (see for instance Section 4.8).

Turn now to the cone \({\mathcal E}\) defined by Equation (3.35). This cone is clearly given by
$${\mathcal E} = \{\lambda \in V\vert \,\forall {\alpha _i}\quad B(\lambda ,{\alpha _i}) > 0\} = \left\{{\sum {{d_i}} {\omega _i}\vert {d_i} > 0} \right\}.$$
(3.39)
Similarly, its closure \({\mathcal F}\) is given by
$${\mathcal F} = \{\lambda \in V\vert \,\forall {\alpha _i}\quad B(\lambda ,{\alpha _i}) \geq 0\} = \left\{{\sum {{d_i}} {\omega _i}\vert {d_i} \geq 0} \right\}.$$
(3.40)
The cone \({\mathcal F}\) is thus the convex hull of the vectors ω i , which are on the boundary of \({\mathcal F}\).

By definition, a hyperbolic Coxeter group is a Lorentzian Coxeter group such that the vectors in \(\mathcal E\) are all timelike, B(λ, λ) < 0 for all \(\lambda \in {\mathcal E}\). Hyperbolic Coxeter groups are precisely the groups that emerge in the gravitational billiards of physical interest. The hyperbolicity condition forces B(λ, λ) < 0 for all \(\lambda \in {\mathcal F}\), and in particular, B(ωi, ω i ) ≤ 0: The fundamental weights are timelike or null. The cone \({\mathcal F}\) then lies within the light cone. This does not occur for generic (non-hyperbolic) Lorentzian algebras.

The following theorem enables one to decide whether a Coxeter group is hyperbolic by mere inspection of its Coxeter graph.

Theorem: Let \({\mathfrak C}\) be a Coxeter group with irreducible Coxeter graph Γ. The Coxeter group is hyperbolic if and only if the following two conditions hold:
  • The bilinear form B is nondegenerate but not positive definite.

  • For each i, the Coxeter graph obtained by removing the node i from Γ is of finite or affine type.

(Note: By removing a node, one might get a non-irreducible diagram even if the original diagram is connected. A reducible diagram defines a Coxeter group of finite type if and only if each irreducible component is of finite type, and a Coxeter group of affine type if and only if each irreducible component is of finite or affine type with at least one component of affine type.)

Proof:
  • It is clear that if a Coxeter group is hyperbolic, then its bilinear form fulfills the first condition. Let ω i be one of the vectors of the dual basis. The vectors α j with ji form a basis of the hyperplane Π i orthogonal to ω i . Because ω i is non-spacelike (the group is hyperbolic), the hyperplane Π i is spacelike or null. The Coxeter graph defined by the α j with ji (i.e., by removing the node α i ) is thus of finite or affine type.

  • Conversely, assume that the two conditions of the theorem hold. From the first condition, it follows that the set N = {λ ∈ V | B(λ, λ) < 0} is non-empty. Let Π i be the hyperplane spanned by the α j with ji, i.e., orthogonal to ω i . From the second condition, it follows that the intersection of N with each Π i is empty. Accordingly, each connected component of N lies in one of the connected components of the complement of \(\bigcup\nolimits_i {{\Pi _i}}\) Π i , namely, is on a definite (positive or negative) side of each of the hyperplanes Π i . These sets are of the form ∈ i c i α i with c i > 0 for some i’s (fixed throughout the set) and c i < 0 for the others. This forces the signature of B to be Lorentzian since otherwise there would be at least a two-dimensional subspace Z of V such that Z {0} ⊂ N. Because Z {0} is connected, it must lie in one of the subsets just described. But this is impossible since if λ ∈ Z {0}, then −λ ∈ Z {0}.

We now show that \({\mathcal E} \subset N\). Because the signature of B is Lorentzian, N is the inside of the standard light cone and has two components, the “future” component and the “past” component. From the second condition of the theorem, each ω i lies on or inside the light cone since the orthogonal hyperplane is non-timelike. Furthermore, all the ω i ’s are future pointing, which implies that the cone \({\mathcal E}\) lies in N, as had to be shown (a positive sum of future pointing non spacelike vectors is non-spacelike). This concludes the proof of the theorem.

In particular, this theorem is useful for determining all hyperbolic Coxeter groups once one knows the list of all finite and affine ones. To illustrate its power, consider the Coxeter diagram of Figure 8, with 8 nodes on the loop and one extra node attached to it (we shall see later that it is called \(A_7^{+ +}\)).
Figure 8

The Coxeter graph of the group \(A_7^{+ +}\).

The bilinear form is given by
$${1 \over 2}\left({\begin{array}{*{20}c} 2 & {- 1} & 0 & 0 & 0 & 0 & 0 & {- 1} & 0 \\ {- 1} & 2 & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {- 1} & 2 & {- 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {- 1} & 2 & {- 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {- 1} & 2 & {- 1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {- 1} & 2 & {- 1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {- 1} & 2 & {- 1} & 0 \\ {- 1} & 0 & 0 & 0 & 0 & 0 & {- 1} & 2 & {- 1} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {- 1} & 2 \\ \end{array}} \right).$$
(3.41)
and is of Lorentzian signature. If one removes the node labelled 9, one gets the affine diagram \(A_7^{+}\) (see Figure 9). If one removes the node labelled 8, one gets the finite diagram of the direct product group A1 × A7 (see Figure 10). Deleting the nodes labelled 1 or 7 yields the finite diagram of A8(see Figure 11). Removing the nodes labelled 2 or 6 gives the finite diagram of D8 (see Figure 12). If one removes the nodes labelled 3 or 5, one obtains the finite diagram of E8 (see Figure 13). Finally, deleting the node labelled 4 yields the affine diagram of \(E_7^{+}\) (see Figure 14). Hence, the Coxeter group is hyperbolic.
Figure 9

The Coxeter graph of \(A_7^{+ +}\).

Figure 10

The Coxeter graph of A7 × A1.

Figure 11

The Coxeter graph of A 8 .

Figure 12

The Coxeter graph of D8.

Figure 13

The Coxeter graph of E8.

Figure 14

The Coxeter graph of \(A_7^{+}\).

Consider now the same diagram, with one more node in the loop \((A_8^{+ +})\). In that case, if one removes one of the middle nodes 4 or 5, one gets the Coxeter group \(E_7^{+ +}\), which is neither finite nor affine. Hence, \(A_8^{+ +}\) is not hyperbolic.

Using the two conditions in the theorem, one can in fact provide the list of all irreducible hyperbolic Coxeter groups. The striking fact about this classification is that hyperbolic Coxeter groups exist only in ranks 3 ≤ n ≤ 10, and, moreover, for 4 ≤ n ≤ 10 there is only a finite number. In the n =3 case, on the other hand, there exists an infinite class of hyperbolic Coxeter groups. In Figure 15 we give a general form of the Coxeter graphs corresponding to all rank 3 hyperbolic Coxeter groups, and in Tables 39 we give the complete classification for 4 ≤ n ≤ 10.
Figure 15

This Coxeter graph corresponds to hyperbolic Coxeter groups for all values of m and n for which the associated bilinear form B is not of positive definite or positive semidefinite type. This therefore gives rise to an infinite class of rank 3 hyperbolic Coxeter groups.

Table 3

Hyperbolic Coxeter groups of rank 4.

Table 4

Hyperbolic Coxeter groups of rank 5.

Table 5

Hyperbolic Coxeter groups of rank 6.

Table 6

Hyperbolic Coxeter groups of rank 7.

Table 7

Hyperbolic Coxeter groups of rank 8.

Table 8

Hyperbolic Coxeter groups of rank 9.

Table 9

Hyperbolic Coxeter groups of rank 10.

Note that the inverse metric (B−1) ij , which gives the scalar products of the fundamental weights, has only negative entries in the hyperbolic case since the scalar product of two future-pointing non-spacelike vectors is strictly negative (it is zero only when the vectors are both null and parallel, which does not occur here).

One can also show [116, 107] that in the hyperbolic case, the Tits cone \({\mathcal X}\) coincides with the future light cone. (In fact, it coincides with either the future light cone or the past light cone. We assume that the time orientation in V has been chosen as in the proof of the theorem, so that the Tits cone coincides with the future light cone.) This is at the origin of an interesting connection with discrete reflection groups in hyperbolic space (which justifies the terminology). One may realize hyperbolic space \({{\mathcal H}_{n - 1}}\) as the upper sheet of the hyperboloid B(λ, λ) = −1 in V. Since the Coxeter group is a subgroup of O+(n − 1,1), it leaves this sheet invariant and defines a group of reflections in \({{\mathcal H}_{n - 1}}\). The fundamental reflections are reflections through the hyperplanes in hyperbolic space obtained by taking the intersection of the Minkowskian hyperplanes B i , λ) = 0 with hyperbolic space. These hyperplanes bound the fundamental region, which is the domain to the positive side of each of these hyperplanes. The fundamental region is a simplex with vertices \({{\bar \omega}_i}\), where \({{\bar \omega}_i}\) are the intersection points of the lines ℝω i with hyperbolic space. This intersection is at infinity in hyperbolic space if ω i is lightlike. The fundamental region has finite volume but is compact only if the ω i are timelike.

Thus, we see that the hyperbolic Coxeter groups are the reflection groups in hyperbolic space with a fundamental domain which (i) is a simplex, and which (ii) has finite volume. The fact that the fundamental domain is a simplex (n vectors in \({{\mathcal H}_{n - 1}}\)) follows from our geometric construction where it is assumed that the n vectors α i form a basis of V.

The group PGL(2, ℤ) relevant to pure gravity in four dimensions is easily verified to be hyperbolic.

For general information, we point out the following facts:
  • Compact hyperbolic Coxeter groups (i.e., hyperbolic Coxeter groups with a compact fundamental region) exist only for ranks 3, 4 and 5, i.e., in two, three and four-dimensional hyperbolic space. All hyperbolic Coxeter groups of rank > 5 have a fundamental region with at least one vertex at infinity. The hyperbolic Coxeter groups appearing in gravitational theories are always of the noncompact type.

  • There exist reflection groups in hyperbolic space whose fundamental domains are not simplices. Amazingly enough, these exist only in hyperbolic spaces of dimension < 995. If one imposes that the fundamental domain be compact, these exist only in hyperbolic spaces of dimension < 29. The bound can probably be improved [164].

  • Non-hyperbolic Lorentzian Coxeter groups are associated through the above construction with infinite-volume fundamental regions since some of the vectors ω i are spacelike, which imply that the corresponding reflection hyperplanes intersect beyond hyperbolic infinity.

3.6 Crystallographic Coxeter groups

Among the Coxeter groups, only those that are crystallographic correspond to Weyl groups of Kac-Moody algebras. Therefore we now introduce this important concept. By definition, a Coxeter group is crystallographic if it stabilizes a lattice in V. This lattice need not be the lattice generated by the α i ’s. As discussed in [107], a Coxeter group is crystallographic if and only if two conditions are satisfied: (i) The integers m ij (i ≠ j) are restricted to be in the set {2, 3, 4, 6, ∞}, and (ii) for any closed circuit in the Coxeter graph of \({\mathfrak C}\), the number of edges labelled 4 or 6 is even.

Given a crystallographic Coxeter group, it is easy to exhibit a lattice L stabilized by it. We can construct a basis for that lattice as follows. The basis vectors μ i of the lattice are multiples of the original simple roots, μ i = c i α i for some scalars c i which we determine by applying the following rules:
  • \({m_{ij}} = 3 \Rightarrow {c_i} = {c_j}\).

  • \({m_{ij}} = 4 \Rightarrow {c_i} = \sqrt 2 {c_j}\,{\rm{or}}\,{c_j} = \sqrt 2 {c_j}\).

  • \({m_{ij}} = 6 \Rightarrow {c_i} = \sqrt 3 {c_j}\,{\rm{or}}\,{c_j} = \sqrt 3 {c_j}\).

  • \({m_{ij}} = \infty \Rightarrow {c_i} = {c_j}\).

One easily verifies that σ i (μ j ) = μ j d ij μ i for some integers d ij . Hence L is indeed stabilized. The integers d ij are equal to \(2{{B({\mu _i},\,{\mu _j})} \over {B({\mu _i},\,{\mu _i})}}\).

The rules are consistent as can be seen by starting from an arbitrary node, say α1, for which one takes c1 = 1. One then proceeds to the next nodes in the (connected) Coxeter graph by applying the above rules. If there is no closed circuit, there is no consistency problem since there is only one way to proceed from α1 to any given node. If there are closed circuits, one must make sure that one comes back to the same vector after one turn around any circuit. This can be arranged if the number of steps where one multiplies or divides by \(\sqrt 2\) (respectively, \(\sqrt 3\)) is even.

Our construction shows that the lattice L is not unique. If there are only two different lengths for the lattice vectors μ i it is convenient to normalize the lengths so that the longest lattice vectors have length squared equal to two. This choice simplifies the factors \(2{{B({\mu _i},\,{\mu _j})} \over {B({\mu _i},\,{\mu _i})}}\).

The rank 10 hyperbolic Coxeter groups are all crystallographic. The lattices preserved by E10 and DE10 are unique up to an overall rescaling because the non-trivial m ij (ij) are all equal to 3 and there is no choice in the ratios c i /c j , all equal to one (first rule above). The Coxeter group BE10 preserves two (dual) lattices.

3.6.1 On the normalization of roots and weights in the crystallographic case

Since the vectors μ i and α i are proportional, they generate identical reflections. Even though they do not necessarily have length squared equal to unity, the vectors μ i are more convenient to work with because the lattice preserved by the Coxeter group is simply the lattice ∈ i ℤμ i of points with integer coordinates in the basis {μ i }. For this reason, we shall call from now on “simple roots” the vectors μ i and, to follow common practice, will sometimes even rename them α i . Thus, in the crystallographic case, the (redefined) simple roots are appropriately normalized to the lattice structure. It turns out that it is with this normalization that simple roots of Coxeter groups correspond to simple roots of Kac-Moody algebras defined in the Section 4.6.3. A root is any point on the root lattice that is in the Coxeter orbit of some (redefined) simple root. It is these roots that coincide with the (real) roots of Kac-Moody algebras.

It is also useful to rescale the fundamental weights. The rescaled fundamental weights, of course proportional to ω i , are denoted Λ i . The convenient normalization is such that
$$({\Lambda _i}\vert {\mu _j}) = {{({\mu _j}\vert {\mu _j})} \over 2}{\delta _{ij}}.$$
(3.42)
With this normalization, they coincide with the fundamental weights of Kac-Moody algebras, to be considered in Section 4.

4 Lorentzian Kac—Moody Algebras

The explicit appearance of infinite crystallographic Coxeter groups in the billiard limit suggests that gravitational theories might be invariant under a huge symmetry described by Lorentzian Kac-Moody algebras (defined in Section 4.1). Indeed, there is an intimate connection between crystallographic Coxeter groups and Kac-Moody algebras. This connection might be familiar in the finite case. For instance, it is well known that the finite symmetry group A2 of the equilateral triangle (isomorphic to the group of permutations of 3 objects) and the corresponding hexagonal pattern of roots are related to the finite-dimensional Lie algebra \(\mathfrak {sl}(3,\,\mathbb R)\) (or \(\mathfrak {su}\)(3)). The group A2 is in fact the Weyl group of \(\mathfrak {sl}(3,\,\mathbb R)\) (see Section 4.7).

This connection is not peculiar to the Coxeter group A2 but is generally valid: Any crystal-lographic Coxeter group is the Weyl group of a Kac-Moody algebra traditionally denoted in the same way (see Section 4.7). This is the reason why it is expected that the Coxeter groups might signal a bigger symmetry structure. And indeed, there are indications that this is so since, as we shall discuss in Section 9, an attempt to reformulate the gravitational Lagrangians in a way that makes the conjectured symmetry manifest yields intriguing results.

The purpose of this section is to develop the mathematical concepts underlying Kac-Moody algebras and to explain the connection between Coxeter groups and Kac-Moody algebras. How this is relevant to gravitational theories will be discussed in Section 5.

4.1 Definitions

An n × n matrix A is called a “generalized Cartan matrix” (or just “Cartan matrix” for short) if it satisfies the following conditions9:
$${A_{ii}} = 2\qquad \forall i = 1, \cdots ,n,$$
(4.1)
$${A_{ij}} \in {{\mathbb Z}_ -}\qquad (i\not = j),$$
(4.1)
$${A_{ij}} = 0\quad \Rightarrow \quad {A_{ji}} = 0,$$
(4.3)
where ℤ_ denotes the non-positive integers. One can encode the Cartan matrix in terms of a Dynkin diagram, which is obtained as follows:
  1. 1.

    For each i = 1, …, n, one associates a node in the diagram.

     
  2. 2.

    One draws a line between the node i and the node j if A ij ≠ 0; if A ij = 0 (= A ij ), one draws no line between i and j.

     
  3. 3.
    One writes the pair (A ij , A ij ) over the line joining i to j. When the products A ij · A ij are all ≤ 4 (which is the only situation we shall meet in practice), this third rule can be replaced by the following rules:
    1. (a)

      one draws a number of lines between i and j equal to max(| A ij |, | A ij |);

       
    2. (b)

      one draws an arrow from j to i if |A ij | > |A ij |.

       
     
So, for instance, the Dynkin diagrams in Figure 16 correspond to the Cartan matrices
$$A[{A_2}] = \left({\begin{array}{*{20}c} 2 & {- 1} \\ {- 1} & 2 \\ \end{array}} \right),$$
(4.4)
$$A[{B_2}] = \left({\begin{array}{*{20}c} 2 & {- 2}\\ {- 1} & 2\\ \end{array}} \right),$$
(4.5)
$$A[{G_2}] = \left({\begin{array}{*{20}c} 2 & {- 3}\\ {- 1} & 2\\ \end{array}} \right),$$
(4.6)
$$A[A_2^{(2)}] = \left({\begin{array}{*{20}c} 2 & {- 4}\\ {- 1} & 2\\ \end{array}} \right),$$
(4.7)
$$A[A_1^ + ] = \left({\begin{array}{*{20}c} 2 & {- 2}\\ {- 2} & 2\\ \end{array}} \right),$$
(4.8)
respectively. If the Dynkin diagram is connected, the matrix A is indecomposable. This is what shall be assumed in the following.
Figure 16

The Dynkin diagrams corresponding to the finite Lie algebras A2, B2 and G2 and to the affine Kac-Moody algebras \(A_2^{(2)}\) and \(A_1^ +\).

Although this is not necessary for developing the general theory, we shall impose two restrictions on the Cartan matrix. The first one is that det A ≠ 0; the second one is that A is symmetrizable. The restriction det A ≠ 0 excludes the important class of affine algebras and will be lifted below. We impose it at first because the technical definition of the Kac-Moody algebra when det A = 0 is then slightly more involved.

The second restriction imposes that there exists an invertible diagonal matrix D with positive elements ϵ i and a symmetric matrix S such that
$$A = DS.$$
(4.9)
The matrix S is called a symmetrization of A and is unique up to an overall positive factor because A is indecomposable. To prove this, choose the first (diagonal) element ϵ1 > 0 of D arbitrarily. Since A is indecomposable, there exists a nonempty set J1 of indices j such that A1j ≠ 0. One has A1j = ϵ1S1j and Aj1 = ϵ j Sj1. This fixes the ϵ j ’s > 0 in terms of ϵ1 since S1j = Sj1. If not all the elements ϵ j are determined at this first step, we pursue the same construction with the elements A jk = ϵ j S jk and A kj = ϵ k S kj = ϵ k S kj with j ϵ J1 and, more generally, at step p, with jJ1J2 ⋯ ∩ J p . As the matrix A is assumed to be indecomposable, all the elements ϵ i of D and S ij of S can be obtained, depending only on the choice of ϵ1. One gets no contradicting values for the ϵ j ’s because the matrix A is assumed to be symmetrizable.

In the symmetrizable case, one can characterize the Cartan matrix according to the signature of (any of) its symmetrization(s). One says that A is of finite type if S is of Euclidean signature, and that it is of Lorentzian type if S is of Lorentzian signature.

Given a Cartan matrix A (with det A ≠ 0), one defines the corresponding Kac-Moody algebra \(\mathfrak {g}=\mathfrak {g}(A)\) as the algebra generated by 3n generators h i , e i , f i subject to the following “Chevalley-Serre” relations (in addition to the Jacobi identity and anti-symmetry of the Lie bracket),
$$\begin{array}{*{20}c} {[{h_i},{h_j}] = 0,\quad \quad} & \\ {[{h_i},{e_j}] = {A_{ij}}{e_j}\quad} & {({\rm{no}}\;{\rm{summation}}\;{\rm{on}}\;j),} \\ {[{h_i},{f_j}] = - {A_{ij}}{f_j}} & {({\rm{no}}\;{\rm{summation}}\;{\rm{on}}\;j),} \\ {[{e_i},{f_j}] = {\delta _{ij}}{h_j}\quad} & {({\rm{no}}\;{\rm{summation}}\;{\rm{on}}\;j),} \\ \end{array}$$
(4.10)
$${\rm{ad}}_{{e_i}}^{1 - {A_{ij}}}({e_j}) = 0,\quad \;\;{\rm{ad}}_{{f_i}}^{1 - {A_{ij}}}({f_j}) = 0,\qquad i\not = j.$$
(4.11)
The relations (4.11), called Serre relations, read explicitly
$$\underbrace {\left[ {{e_i},} \right.\left[ {{e_i},} \right.\left[ {{e_i}, \cdots ,\left[ {{e_i},{e_j}} \right]} \right]\left. \cdots \right] = 0}_{1 - {A_{ij}}\,{\rm{commutators}}}$$
(4.12)
(and likewise for the f k ’s).
Any multicommutator can be reduced, using the Jacobi identity and the above relations, to a multicommutator involving only the e i ’s, or only the f i ’s. Hence, the Kac-Moody algebra splits as a direct sum (“triangular decomposition”)
$$\mathfrak{g} = {\mathfrak{n}_ -} \oplus\mathfrak{h} \oplus {\mathfrak{n}_ +},$$
(4.13)
where \({\mathfrak {n}_ -}\) is the subalgebra involving the multicommutators \([{f_{{i_1}}},\,[{f_{{i_2}}},\, \cdots, \,[{f_{{i_{k - 1}}}},\,{f_{{i_k}}}] \cdots ],\,{\mathfrak {n}_ +}\) is the subalgebra involving the multicommutators \([{e_{{i_1}}},\,[{e_{{i_2}}},\, \cdots, \,[{e_{{i_{k - 1}}}},\,{e_{{i_k}}}] \cdots ]\) and \({\mathfrak h}\) is the Abelian subalgebra containing the h i ’s. This is called the Cartan subalgebra and its dimension n is the rank of the Kac-Moody algebra \({\mathfrak g}\). It should be stressed that the direct sum Equation (4.13) is a direct sum of \({\mathfrak {n}_ -}\), \({\mathfrak h}\) and \({\mathfrak {n}_ +}\) as vector spaces, not as subalgebras (since these subalgebras do not commute).
A priori, the numbers of the multicommutators
$$[{f_{{i_1}}},[{f_{{i_2}}}, \cdots ,[{f_{{i_{k - 1}}}},{f_{{i_k}}}] \cdots ]\qquad {\rm{and}}\qquad [{e_{{i_1}}},[{e_{{i_2}}}, \cdots ,[{e_{{i_{k - 1}}}},{e_{{i_k}}}] \cdots ]$$
are infinite, even after one has taken into account the Jacobi identity. However, the Serre relations impose non-trivial relations among them, which, in some cases, make the Kac-Moody algebra finite-dimensional. Three worked examples are given in Section 4.4 to illustrate the use of the Serre relations. In fact, one can show [116] that the Kac-Moody algebra is finite-dimensional if and only if the symmetrization S of A is positive definite. In that case, the algebra is one of the finite-dimensional simple Lie algebras given by the Cartan classification. The list is given in Table 10.

When the Cartan matrix A is of Lorentzian signature the Kac-Moody algebra \({\mathfrak {g}(A)}\), constructed from A using the Chevalley-Serre relations, is called a Lorentzian Kac-Moody algebra. This is the case of main interest for the remainder of this paper.

4.2 Roots

The adjoint action of the Cartan subalgebra on \({\mathfrak {n}_ +}\) and \({\mathfrak {n}_ -}\) is diagonal. Explicitly,
$$[h,{e_i}] = {\alpha _i}(h){e_i}\qquad ({\rm{no}}\;{\rm{summation}}\;{\rm{on}}\;i)$$
(4.14)
for any element \(h \in {\mathfrak h}\), where α i is the linear form on \({\mathfrak h}\) (i.e., the element of the dual \({\mathfrak h^{\ast}}\)) defined by α i (h j ) = A ji . The α i ’s are called the simple roots. Similarly,
$$[h,[{e_{{i_1}}},[{e_{{i_2}}}, \cdots ,[{e_{{i_{k - 1}}}},{e_{{i_k}}}] \cdots ]] = ({\alpha _{{i_1}}} + {\alpha _{{i_2}}} + \cdots {\alpha _{{i_k}}})(h)\,[{e_{{i_1}}},[{e_{{i_2}}}, \cdots ,[{e_{{i_{k - 1}}}},{e_{{i_k}}}] \cdots ]$$
(4.15)
and, if \([{e_{{i_1}}},\,[{e_{{i_2}}},\, \cdots, \,[{e_{{i_{k - 1}}}},\,{e_{{i_k}}}] \cdots ]\) is non-zero, one says that \({\alpha _{{i_1}}} + {\alpha _{{i_2}}} + \cdots {\alpha _{{i_k}}}\) is a positive root. On the negative side, \({\mathfrak {n}_ -}\), one has
$$[h,[{f_{{i_1}}},[{f_{{i_2}}}, \cdots ,[{f_{{i_{k - 1}}}},{f_{{i_k}}}] \cdots ]] = - ({\alpha _{{i_1}}} + {\alpha _{{i_2}}} + \cdots {\alpha _{{i_k}}})(h)\,[{f_{{i_1}}},[{f_{{i_2}}}, \cdots ,[{f_{{i_{k - 1}}}},{f_{{i_k}}}] \cdots ]$$
(4.16)
and \(- ({\alpha _{{i_1}}} + {\alpha _{{i_2}}} + \cdots {\alpha _{{i_k}}})(h)\) is called a negative root when \([{f_{{i_1}}},\,[{f_{{i_2}}},\, \cdots, \,[{f_{{i_{k - 1}}}},\,{f_{{i_k}}}]\) is non-zero. This occurs if and only if \([{e_{{i_1}}},[{e_{{i_2}}}, \cdots ,[{e_{{i_{k - 1}}}},{e_{{i_k}}}] \cdots ]\) is non-zero: −α is a negative root if and only if α is a positive root.
We see from the construction that the roots (linear forms α such that [h, x] = α(h)x has nonzero solutions x) are either positive (linear combinations of the simple roots α i with integer non-negative coefficients) or negative (linear combinations of the simple roots with integer non-positive coefficients). The set of positive roots is denoted by Δ+; that of negative roots by Δ. The set of all roots is Δ, so we have Δ = Δ+ ∪Δ. The simple roots are positive and form a basis of \({\mathfrak h^{\ast}}\). One sometimes denotes the h i by \(\alpha _i^ \vee\) (and thus, \([\alpha _i^ \vee, \,{e_j}] = {A_{ij}}{e_j}\) etc). Similarly, one also uses the notation 〈·,·〉 for the standard pairing between \({\mathfrak h}\) and its dual \({\mathfrak h^{\ast}}\), i.e., 〈α, h〉 = α(h). In this notation the entries of the Cartan matrix can be written as
$${A_{ij}} = {\alpha _j}(\alpha _i^ \vee) = \langle {\alpha _j},\alpha _i^ \vee \rangle {.}$$
(4.17)
Finally, the root lattice Q is the set of linear combinations with integer coefficients of the simple roots,
$$Q = \sum\limits_i {{\mathbb Z}\alpha _i}.$$
(4.18)

All roots belong to the root lattice, of course, but the converse is not true: There are elements of Q that are not roots.

4.3 The Chevalley involution

The symmetry between the positive and negative subalgebras \({\mathfrak {n}_ +}\) and \({\mathfrak {n}_ -}\) of the Kac-Moody algebra can be rephrased formally as follows: The Kac-Moody algebra is invariant under the Chevalley involution τ, defined on the generators as
$$\tau ({h_i}) = - {h_i},\qquad \tau ({e_i}) = - {f_i},\qquad \tau ({f_i}) = - {e_i}.$$
(4.19)
The Chevalley involution is in fact an algebra automorphism that exchanges the positive and negative sides of the algebra.

Finally, we quote the following useful theorem.

Theorem: The Kac-Moody algebra \({\mathfrak g}\) defined by the relations (4.10, 4.11) is simple. The proof may be found in Kac’ book [116], page 12.

We note that invertibility and indecomposability of the Cartan matrix A are central ingredients in the proof. In particular, the theorem does not hold in the affine case, for which the Cartan matrix is degenerate and has non-trivial ideals10 (see [116] and Section 4.5).

4.4 Three examples

To get a feeling for how the Serre relations work, we treat in detail three examples.
  • A2: We start with A2, the Cartan matrix of which is Equation (4.4). The defining relations are then:
    $$\begin{array}{*{20}c} {[{h_1},{h_2}] = 0,\quad \;\;} & {[{h_1},{e_1}] = 2{e_1},\;\;} & {[{h_1},{e_2}] = - {e_2},\quad} \\ {[{h_1},{f_1}] = - 2{f_1},} & {[{h_1},{f_2}] = {f_2},\quad} & {[{h_2},{e_1}] = - {e_1},\quad} \\ {[{h_2},{e_2}] = 2{e_2},\;\;} & {[{h_2},{f_1}] = {f_1},\quad} & {[{h_2},{f_2}] = - 2{f_2},\;\;} \\ {[{e_1},[{e_1},{e_2}]] = 0,\;\;\quad \quad \;\;} & {[{e_2},[{e_2},{e_1}]] = 0,\quad \quad \quad} & {[{f_1},[{f_1},{f_2}]] = 0,\quad \quad \quad \quad} \\ {[{f_2},[{f_2},{f_1}]] = 0\;\;\quad \quad \quad \;} & {[{e_i},{f_j}] = {\delta _{ij}}{h_j}.} & {} \\ \end{array}$$
    (4.20)
    The commutator [e1, e2] is not killed by the defining relations and hence is not equal to zero (the defining relations are all the relations). All the commutators with three (or more) e’s are however zero. A similar phenomenon occurs on the negative side. Hence, the algebra A2 is eight-dimensional and one may take as basis {h1, h2, e1, e2, [e1, e2], f1, f2, [f1, f2]}. The vector [e1, e2] corresponds to the positive root α1 + α2.
  • B2: The algebra B2, the Cartan matrix of which is Equation (4.5), is defined by the same set of generators, but the Serre relations are now [e1, [e1, [e1, e2]]] = 0 and [e2, [e2, e1]] = 0 (and similar relations for the f’s). The algebra is still finite-dimensional and contains, besides the generators, the commutators [e1, e2], [e1, [e1, e2]], their negative counterparts [f1, f2] and [f1, [f1, f2]], and nothing else. The triple commutator [e1, [e1, [e1, e2]]] vanishes by the Serre relations. The other triple commutator [e2, [e1, [e1, e2]]] vanishes also by the Jacobi identity and the Serre relations,
    $$[{e_1},[{e_1},[{e_1},{e_2}]]] = [[{e_2},{e_1}],[{e_1},{e_2}]] + [{e_1},[{e_2},[{e_1},{e_2}]]] = 0.$$
    (Each term on the right-hand side is zero: The first by antisymmetry of the bracket and the second because [e2, [e1, e2]] = −[e2, [e2, e1]] = 0.) The algebra is 10-dimensional and is isomorphic to \({\mathfrak {so}(3,\,2)}\).
  • \(A_1^{+}\): We now turn to \(A_1^{+}\), the Cartan matrix of which is Equation (4.8). This algebra is defined by the same set of generators as A2, but with Serre relations given by
    $$\begin{array}{*{20}c} {[{e_1},[{e_1},[{e_1},{e_2}]]] = 0,}\\ {[{e_2},[{e_2},[{e_2},{e_1}]]] = 0}\\\end{array}$$
    (4.21)
    (and similar relations for the f’s). This innocent-looking change in the Serre relations has dramatic consequences because the corresponding algebra is infinite-dimensional. (We analyze here the algebra generated by the h’s, e’s and f’s, which is in fact the derived Kac-Moody algebra — see Section 4.5 on affine Kac-Moody algebras. The derived algebra is already infinite-dimensional.) To see this, consider the \({\mathfrak {sl}(2,\,\mathbb R)}\) current algebra, defined by
    $$[J_m^a,J_n^b] = {f^{ab}}_cJ_{m + n}^c + m{k^{ab}}c{\delta _{m + n,0}},$$
    (4.22)
    where \(a = 3,\, +, \, -, \,{f^{ab}}_c\) are the structure constants of \({\mathfrak {sl}(2,\,\mathbb R)}\) and where k ab is the invariant metric on \({\mathfrak {sl}(2,\,\mathbb R)}\) which we normalize here so that k−+ = 1. The subalgebra with n = 0 is isomorphic to \({\mathfrak {sl}(2,\,\mathbb R)}\),
    $$[J_0^3,J_0^ + ] = 2J_0^ + ,\qquad [J_0^3,J_0^ - ] = - 2J_0^ - ,\qquad [J_0^ + ,J_0^ - ] = J_0^3.$$
    The current algebra (4.22) is generated by \(J_0^a\), c, \(J_0^ -\) and \(J_{- 1}^ +\) since any element can be written as a multi-commutator involving them. The map
    $$\begin{array}{*{20}c} {{h_1}} & \rightarrow & {J_0^3,} & {{h_2}} & \rightarrow & {- J_0^3 + c,} \\ {{e_1}} & \rightarrow & {J_0^ + ,} & {{e_2}} & \rightarrow & {J_1^ - ,} \\ {{f_1}} & \rightarrow & {J_0^ - ,} & {{f_2}} & \rightarrow & {J_{- 1}^ +} \\ \end{array}$$
    (4.23)
    preserves the defining relations of the Kac-Moody algebra and defines an isomorphism of the (derived) Kac-Moody algebra with the current algebra. The Kac-Moody algebra is therefore infinite-dimensional. One can construct non-vanishing infinite multi-commutators, in which e1 and e2 alternate:
    $$\begin{array}{*{20}c} {\left[ {{e_1},} \right.\left[ {{e_2},} \right.\left[ {{e_1}, \cdots ,\left[ {{e_1},{e_2}} \right]\left. {\left. \cdots \right]} \right]} \right]\sim J_n^3\;\;\;} & {(n\;{e_1}\prime{\rm{s}}\;{\rm{and}}\;n\;{e_2}\prime{\rm{s}}),\;\;\;\;} \\ {\left[ {{e_1},} \right.\left[ {{e_2},} \right.\left[ {{e_1}, \cdots ,\left[ {{e_2},{e_1}} \right]\left. {\left. \cdots \right]} \right]} \right]\sim J_n^ + \;\;} & {(n + 1\;{e_1}\prime{\rm{s}}\;{\rm{and}}\;n\;{e_2}\prime{\rm{s}}),\;} \\ {\left[ {{e_2},} \right.\left[ {{e_1},} \right.\left[ {{e_2}, \cdots ,\left[ {{e_1},{e_2}} \right]\left. {\left. \cdots \right]} \right]} \right]\sim J_{n + 1}^ -} & {(n\;{e_1}\prime{\rm{s}}\;{\rm{and}}\;n + 1\;{e_2}\prime{\rm{s}}).} \\ \end{array}$$
    (4.24)
    The Serre relations do not cut the chains of multi-commutators to a finite number.
We see from these examples that the exact consequences of the Serre relations might be intricate to derive explicitly. This is one of the difficulties of the theory.

4.5 The affine case

The affine case is characterized by the conditions that the Cartan matrix has vanishing determinant, is symmetrizable and is such that its symmetrization S is positive semi-definite (only one zero eigenvalue). As before, we also take the Cartan matrix to be indecomposable. By a reasoning analogous to what we did in Section 3.4 above, one can show that the radical of S is one-dimensional and that the ranks of S and A are equal to n − 1.

One defines the corresponding Kac-Moody algebras in terms of 3n + 1 generators, which are the same generators h i , e i , f i subject to the same conditions (4.10, 4.11) as above, plus one extra generator η which can be taken to fulfill
$$[\eta ,{h_i}] = 0,\qquad [\eta ,{e_i}] = {\delta _{1i}}{e_1},\qquad [\eta ,{f_i}] = - {\delta _{1i}}{f_1}.$$
(4.25)
The algebra admits the same triangular decomposition as above,
$$\mathfrak{g} = {\mathfrak{n}_ -} \oplus\mathfrak{h} \oplus {\mathfrak{n}_ +},$$
(4.26)
but now the Cartan subalgebra \({\mathfrak h}\) has dimension n + 1 (it contains the extra generator n).
Because the matrix A ij has vanishing determinant, one can find a i such that ∑ i a i A ij = 0. The element c = ∑ i a i h i is in the center of the algebra. In fact, the center of the Kac-Moody algebra is one-dimensional and coincides with ℂc [116]. The derived algebra \({\mathfrak {g}{\prime}} = [\mathfrak {g},\,\mathfrak {g}]\) is the subalgebra generated by h i , e i , f i and has codimension one (it does not contain η). One has
$$\mathfrak{g}=\mathfrak{g} \prime \oplus {\mathbb C}\eta$$
(4.27)
(direct sum of vector spaces, not as algebras). The only proper ideals of the affine Kac-Moody algebra \({\mathfrak g}\) are \({\mathfrak {g}{\prime}}\) and ℂc.
Affine Kac-Moody algebras appear in the BKL context as subalgebras of the relevant Lorentzian Kac-Moody algebras. Their complete list is known and is given in Tables 11 and 12.
Table 11

Untwisted affine Kac-Moody algebras.

Table 12

Twisted affine Kac-Moody algebras. We use the notation of Kac [116].

Name

Dynkin diagram

\(A_2^{(2)}\)

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\(A_{2n}^{(2)}\,(n \geq 2)\)

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\(A_{2n - 1}^{(2)}\,(n \geq 3)\)

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\(D_{n + 1}^{(2)}\)

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\(E_6^{(2)}\)

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\(D_4^{(3)}\)

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4.6 The invariant bilinear form

4.6.1 Definition

To proceed, we assume, as announced above, that the Cartan matrix is invertible and symmetrizable since these are the only cases encountered in the billiards. Under these assumptions, an invertible, invariant bilinear form is easily defined on the algebra. We denote by ϵ i the diagonal elements of D,
$$A = DS,\qquad D = {\rm{diag}}({\epsilon_1},{\epsilon_2} \cdots ,{\epsilon_n}){.}$$
(4.28)
First, one defines an invertible bilinear form in the dual \({\mathfrak h^{\ast}}\) of the Cartan subalgebra. This is done by simply setting
$$({\alpha _i}\vert {\alpha _j}) = {S_{ij}}$$
(4.29)
for the simple roots. It follows from A ii = 2 that
$${\epsilon_i} = {2 \over {({\alpha _i}\vert {\alpha _i})}}$$
(4.30)
and thus the Cartan matrix can be written as
$${A_{ij}} = 2{{({\alpha _i}\vert {\alpha _j})} \over {({\alpha _i}\vert {\alpha _i})}}.$$
(4.31)
It is customary to fix the normalization of S so that the longest roots have (α i |α i ) = 2. As we shall now see, the definition (4.29) leads to an invariant bilinear form on the Kac-Moody algebra.
Since the bilinear form (·|·) is nondegenerate on \({\mathfrak h^{\ast}}\), one has an isomorphism \(\mu : \mathfrak {h}^{\ast} \rightarrow \mathfrak {h}\) defined by
$$\langle \alpha ,\mu (\gamma)\rangle = (\alpha \vert \gamma){.}$$
(4.32)
This isomorphism induces a bilinear form on the Cartan subalgebra, also denoted by (·|·). The inverse isomorphism is denoted by v and is such that
$$\langle \nu (h),h\prime \rangle = (h\vert h\prime),\qquad h,h\prime \in \mathfrak{h}.$$
(4.33)
Since the Cartan elements \({h_i} \equiv \alpha _i^ \vee\) obey
$$\langle {\alpha _i},\alpha _j^ \vee \rangle = {A_{ji}},$$
(4.34)
it is clear from the definitions that
$$\nu ({h_i}) \equiv \nu (\alpha _i^ \vee) = {\epsilon_i}{\alpha _i}\quad \Leftrightarrow \quad {h_i} \equiv \alpha _i^ \vee = {{2\mu ({\alpha _i})} \over {({\alpha _i}\vert {\alpha _i})}},$$
(4.35)
and thus also
$$({h_i}\vert {h_j}) = {\epsilon_i}{\epsilon_j}{S_{ij}}.$$
(4.36)
The bilinear form (·|·) can be uniquely extended from the Cartan subalgebra to the entire algebra by requiring that it is invariant, i.e., that it fulfills
$$([x,y]\vert z) = (x\vert [y,z])\quad \forall \,x,y,z \in \mathfrak{g}.$$
(4.37)
for instance, for the e i ’s and f i ’s one finds
$$({h_i}\vert {e_j}){A_{kj}} = ({h_i}\vert [{h_k},{e_j}]) = ([{h_i},{h_k}]\vert {e_j}) = 0\quad \Rightarrow \quad ({h_i}\vert {e_j}) = 0,$$
(4.38)
and similarly
$$({h_i}\vert {f_j}) = 0{.}$$
(4.39)
In the same way we have
$${A_{ij}}({e_j}\vert {f_k}) = ([{h_i},{e_j}]\vert {f_k}) = ({h_i}\vert [{e_j},{f_k}]) = ({h_i}\vert {h_j}){\delta _{jk}} = {A_{ij}}{\epsilon_j}{\delta _{jk}},$$
(4.40)
and thus
$$({e_i}\vert {f_j}) = {\epsilon_i}{\delta _{ij}}.$$
(4.41)
Quite generally, if e α and e γ are root vectors corresponding respectively to the roots α and γ,
$$[h,{e_\alpha}] = \alpha (h){e_\alpha},\qquad [h,{e_\gamma}] = \gamma (h){e_\gamma},$$
then (e α |e γ ) = 0 unless γ = −α. Indeed, one has
$$\alpha (h)({e_\alpha}\vert {e_\gamma}) = ([h,{e_\alpha}]\vert {e_\gamma}) = - ({e_\alpha}\vert [h,{e_\gamma}]) = - \gamma (h)({e_\alpha}\vert {e_\gamma}),$$
and thus
$$({e_\alpha}\vert {e_\gamma}) = 0\qquad {\rm{if}}\;\alpha + \gamma \not = 0.$$
(4.42)
It is proven in [116] that the invariance condition on the bilinear form defines it indeed consistently and that it is nondegenerate. Furthermore, one finds the relations
$$[h,x] = \alpha (h)x,\qquad [h,y] = - \alpha (h)y\quad \Rightarrow \quad [x,y] = (x\vert y)\mu (\alpha){.}$$
(4.43)

4.6.2 Real and imaginary roots

Consider the restriction (·|·) of the bilinear form to the real vector space \(\mathfrak {h} _{\mathbb {R}}^ {\ast}\) obtained by taking the real span of the simple roots,
$$\mathfrak{h}_{\mathbb R}^ \star = \sum\limits_i {\mathbb R}\alpha_{i}.$$
(4.44)
This defines a scalar product with a definite signature. As we have mentioned, the signature is Euclidean if and only if the algebra is finite-dimensional [116]. In that case, all roots — and not just the simple ones — are spacelike, i.e., such that (α|α) > 0.

When the algebra is infinite-dimensional, the invariant scalar product does not have Euclidean signature. The spacelike roots are called “real roots”, the non-spacelike ones are called “imaginary roots” [116]. While the real roots are nondegenerate (i.e., the corresponding eigenspaces, called “root spaces”, are one-dimensional), this is not so for imaginary roots. In fact, it is a challenge to understand the degeneracy of imaginary roots for general indefinite Kac-Moody algebras, and, in particular, for Lorentzian Kac-Moody algebras.

Another characteristic feature of real roots, familiar from standard finite-dimensional Lie algebra theory, is that if α is a (real) root, no multiple of α is a root except ±α. This is not so for imaginary roots, where 2α (or other non-trivial multiples of α) can be a root even if α is. We shall provide explicit examples below.

Finally, while there are at most two different root lengths in the finite-dimensional case, this is no longer true even for real roots in the case of infinite-dimensional Kac-Moody algebras11. When all the real roots have the same length, one says that the algebra is “simply-laced”. Note that the imaginary roots (if any) do not have the same length, except in the affine case where they all have length squared equal to zero.

4.6.3 Fundamental weights and the Weyl vector

The fundamental weights {Λ i } of the Kac-Moody algebra are vectors in the dual space \({\mathfrak h^{\ast}}\) of the Cartan subalgebra defined by
$$\langle {\Lambda _i},\alpha _j^ \vee \rangle = {\delta _{ij}}.$$
(4.45)
This implies
$$({\Lambda _i}\vert {\alpha _j}) = {{{\delta _{ij}}} \over {{\epsilon_j}}}.$$
(4.46)
The Weyl vector ρ\(\rho \in {\mathfrak h^{\ast}}\) is defined by
$$(\rho \vert {\alpha _j}) = {1 \over {{\epsilon_j}}}$$
(4.47)
and is thus equal to
$$\rho = \sum\limits_i {{\Lambda _i}} .$$
(4.48)

4.6.4 The generalized Casimir operator

From the invariant bilinear form, one can construct a generalized Casimir operator as follows.

We denote the eigenspace associated with α by \({\mathfrak {g}_\alpha}\). This is called the “root space” of α and is defined as
$${\mathfrak{g}_\alpha} = \{x \in \mathfrak{g}\,\vert \,[x,h] = \alpha (h)x,\quad \forall \,h \in\mathfrak{h} \}{.}$$
(4.49)
A representation of the Kac-Moody algebra is called restricted if for every vector v of the representation subspace V, one has \({\mathfrak {g}_\alpha \cdot \upsilon = 0}\) for all but a finite number of positive roots α.
Let \(\{e_\alpha ^K\}\) be a basis of \({\mathfrak {g}_\alpha}\) and let \(\{e_{- \alpha}^K\}\) be the basis of \({\mathfrak {g}_ {- {\alpha}}}\) dual to \(\{e_\alpha ^K\}\) in the B-metric,
$$(e_\alpha ^K\vert e_{- \alpha}^L) = {\delta ^{KL}}.$$
(4.50)
Similarly, let {u i } be a basis of \({\mathfrak {g}_\alpha}\) and {u i } the dual basis of \({\mathfrak h}\) with respect to the bilinear form
$$({u_i}\vert {u^j}) = \delta _i^j.$$
(4.51)
We set
$$\Omega = 2\mu (\rho) + \sum\limits_i {{u^i}} {u_i} + 2\sum\limits_{\alpha \in {\Delta _ +}} {\sum\limits_K {e_{- \alpha}^K}} e_\alpha ^K,$$
(4.52)
where ρ is the Weyl vector. Recall from Section 4.6.1 that μ is an isomorphism from \({\mathfrak h^{\star}}\) to \({\mathfrak h}\), so, since ρ ∈ \(\rho \in {\mathfrak h^{\star}}\), the expression μ(ρ) belongs to \({\mathfrak h}\) as required. When acting on any vector of a restricted representation, Ω is well-defined since only a finite number of terms are different from zero.

It is proven in [116] that Ω commutes with all the operators of any restricted representation. For that reason, it is known as the (generalized) Casimir operator. It is quadratic in the generators12.

4.6.4.1 Note
This definition — and, in particular, the presence of the linear term μ(ρ) — might seem a bit strange at first sight. To appreciate it, turn to a finite-dimensional simple Lie algebra. In the above notations, the usual expression for the quadratic Casimir operator reads
$${\Omega _{{\rm{finite}}}} = \sum\limits_A {{\kappa ^{AB}}} {T_A}{T_B} = \sum\limits_i {{u^i}} {u_i} + \sum\limits_{\alpha \in {\Delta _ +}} {({e_{- \alpha}}{e_\alpha} + {e_\alpha}{e_{- \alpha}})}$$
(4.53)
(without degeneracy index K since the roots are nondegenerate in the finite-dimensional case). Here, κ AB is the Killing metric and {T A } a basis of the Lie algebra. This expression is not “normal-ordered” because there are, in the last term, lowering operators standing on the right. We thus replace the last term by
$$\begin{array}{*{20}c} {\sum\limits_{\alpha \in {\Delta _ +}} {{e_\alpha}} {e_{- \alpha}} = \sum\limits_{\alpha \in {\Delta _ +}} {{e_{- \alpha}}} {e_\alpha} + \sum\limits_{\alpha \in {\Delta _ +}} {[{e_\alpha},{e_{- \alpha}}]\quad \quad \;\;\;}} \\ {= \sum\limits_{\alpha \in {\Delta _ +}} {{e_{- \alpha}}} {e_\alpha} + \sum\limits_{\alpha \in {\Delta _ +}} \mu (\alpha){.}} \\\end{array}$$
(4.54)
Using the fact that in a finite-dimensional Lie algebra, \(\rho = \left({1/2} \right)\sum\nolimits_{\alpha \in {\Delta _ +}} \alpha\), (see, e.g., [85]) one sees that the Casimir operator can be rewritten in “normal ordered” form as in Equation (4.52). The advantage of the normal-ordered form is that it makes sense also for infinite-dimensional Kac-Moody algebras in the case of restricted representations.

4.7 The Weyl group

The Weyl group \(\mathfrak{W}[\mathfrak{g}]\) of a Kac-Moody algebra \(\mathfrak{g}\) is a discrete group of transformations acting on \(\mathfrak{h}^{\ast}\). It is defined as follows. One associates a “fundamental Weyl reflection” \({r_i} \in \mathfrak{W}[\mathfrak{g}]\) to each simple root through the formula
$${r_i}(\lambda) = \lambda - 2{{(\lambda \vert {\alpha _i})} \over {({\alpha _i}\vert {\alpha _i})}}{\alpha _i}.$$
(4.55)
The Weyl group is just the group generated by the fundamental Weyl reflections. In particular,
$${r_i}({\alpha _j}) = {\alpha _j} - {A_{ij}}{\alpha _i}\quad \;\;({\rm{no}}\;{\rm{summation}}\;{\rm{on}}\;i){.}$$
(4.56)
The Weyl group enjoys a number of interesting properties [116]:
  • It preserves the scalar product on \(\mathfrak{h}^{\ast}\).

  • It preserves the root lattice and hence is crystallographic.

  • Two roots that are in the same orbit have identical multiplicities.

  • Any real root has in its orbit (at least) one simple root and hence, is nondegenerate.

  • The Weyl group is a Coxeter group. The connection between the Coxeter exponents and the Cartan integers A ij is given in Table 13 (ij).

This close relationship between Coxeter groups and Kac-Moody algebras is the reason for denoting both with the same notation (for instance, A n denotes at the same time the Coxeter group with Coxeter graph of type A n and the Kac-Moody algebra with Dynkin diagram A n ).
Table 13

Cartan integers and Coxeter exponents.

A ij A ji

m ij

0

2

1

3

2

4

3

6

Note that different Kac-Moody algebras may have the same Weyl group. This is in fact already true for finite-dimensional Lie algebras, where dual algebras (obtained by reversing the arrows in the Dynkin diagram) have the same Weyl group. This property can be seen from the fact that the Coxeter exponents are related to the duality-invariant product A ij A ji . But, on top of this, one sees that whenever the product A ij A ji exceeds four, which occurs only in the infinite-dimensional case, the Coxeter exponent m ij is equal to infinity, independently of the exact value of A ij A ji . Information is thus clearly lost. For example, the Cartan matrices
$$\left({\begin{array}{*{20}c} 2 & {- 2} & {- 2} \\ {- 2} & 2 & {- 2} \\ {- 2} & {- 2} & 2 \\ \end{array}} \right),\qquad \left({\begin{array}{*{20}c} 2 & {- 9} & {- 8} \\ {- 4} & 2 & {- 5} \\ {- 3} & {- 7} & 2 \\\end{array}} \right)$$
(4.57)
lead to the same Weyl group, even though the corresponding Kac-Moody algebras are not isomorphic or even dual to each other.

Because the Weyl groups are (crystallographic) Coxeter groups, we can use the theory of Coxeter groups to analyze them. In the Kac-Moody context, the fundamental region is called “the fundamental Weyl chamber”.

We also note that by (standard vector space) duality, one can define the action of the Weyl group in the Cartan subalgebra \(\mathfrak{h}\), such that
$$\langle \gamma ,r_i^ \vee (h)\rangle = \langle {r_i}(\gamma),h\rangle \qquad {\rm{for}}\;\gamma \in {\mathfrak{h}^ \star}{\rm{and}}\;h \in\mathfrak{h}.$$
(4.58)
One has using Equations (4.30, 4.32, 4.33, 4.35),
$$r_i^ \vee (h) = h - \langle {\alpha _i},h\rangle {h_i} = h - 2{{(h\vert {h_i})} \over {({h_i}\vert {h_i})}}{h_i}{.}$$
(4.59)

Finally, we leave it to the reader to verify that when the products A ij A ji are all ≤ 4, then the geometric action of the Coxeter group considered in Section 3.2.4 and the geometric action of the Weyl group considered here coincide. The (real) roots and the fundamental weights differ only in the normalization and, once this is taken into account, the metrics coincide. This is not the case when some products A ij A ji exceed 4. It should be also pointed out that the imaginary roots of the Kac-Moody algebras do not have immediate analogs on the Coxeter side.

4.7.1 Examples

  • Consider the Cartan matrices
    $$A\prime = \left({\begin{array}{*{20}c} 2 & {- 2} & 0 \\ {- 2} & 2 & {- 1} \\ 0 & {- 1} & 2 \\ \end{array}} \right),\qquad A\prime \prime = \left({\begin{array}{*{20}c} 2 & {- 4} & 0 \\ {- 1} & 2 & {- 1} \\ 0 & {- 1} & 2 \\ \end{array}} \right)$$
    As the first (respectively, second) Cartan matrix defines the Lie algebra \(A_1^{+ +}\) (respectively \(A_2^{(2) +}\)) introduced below in Section 4.9, we also write it as \(A\prime \equiv A[A_1^{+ +}]\) (respectively, \(A\prime \prime \equiv A[A_2^{(2) +}]\)). We denote the associated sets of simple roots by {α1, α2, α3} and {α1, α2, α3}, respectively. In both cases, the Coxeter exponents are m12 = ∞, m13 = 2, m23 = 3 and the metric B ij of the geometric Coxeter construction is
    $$A\prime = \left({\begin{array}{*{20}c} 1 & {- 1} & 0 \\ {- 1} & 1 & {- {1 \over 2}} \\ 0 & {- {1 \over 2}} & 1 \\ \end{array}} \right).$$
    We associate the simple roots {α1, α2, α3} with the geometric realisation of the Coxeter group \(\mathfrak{B}\) defined by the matrix B. These roots may a priori differ by normalizations from the simple roots of the Kac-Moody algebras described by the Cartan matrices A′ and A″.
    Choosing the longest Kac-Moody roots to have squared length equal to two yields the scalar products
    $$S\prime = \left({\begin{array}{*{20}c} 2 & {- 2} & 0 \\ {- 2} & 2 & {- 1} \\ 0 & {- 1} & 2 \\ \end{array}} \right),\qquad S\prime \prime = \left({\begin{array}{*{20}c} {{1 \over 2}} & {- 1} & 0 \\ {- 1} & 2 & {- 1} \\ 0 & {- 1} & 2 \\ \end{array}} \right).$$
    Recall now from Section 3 that the fundamental reflections \({\sigma _i} \in \mathfrak{B}\) have the following geometric realisation
    $${\sigma _i}({\alpha _j}) = {\alpha _j} - 2{B_{ij}}{\alpha _i}\qquad (i = 1,2,3),$$
    (4.60)
    which in this case becomes
    $$\begin{array}{*{20}c} {{\sigma _1}:} & {{\alpha _1} \rightarrow - {\alpha _1},\quad \quad} & {{\alpha _2} \rightarrow {\alpha _2} + 2{\alpha _1},\;} & {{\alpha _3} \rightarrow {\alpha _3},\quad \quad} \\ {{\sigma _2}:} & {{\alpha _1} \rightarrow {\alpha _1} + 2{\alpha _2},} & {{\alpha _2} \rightarrow - {\alpha _2},\quad \quad} & {{\alpha _3} \rightarrow {\alpha _3} + {\alpha _2},} \\ {{\sigma _3}:} & {{\alpha _1} \rightarrow {\alpha _1},\quad \quad} & {{\alpha _2} \rightarrow {\alpha _2} + {\alpha _3},\;} & {{\alpha _3} \rightarrow - {\alpha _3}.\quad \;} \\\end{array}$$
    We now want to compare this geometric realisation of \(\mathfrak{B}\) with the action of the Weyl groups of A′ and A″ on the corresponding simple roots α′ i and α″ i . According to Equation (4.56), the Weyl group \(\mathfrak{W}[A_1^{+ +}]\) acts as follows on the roots α′ i
    $$\begin{array}{*{20}c} {{{r\prime}_1}:} & {{{\alpha \prime}_1} \rightarrow - {{\alpha \prime}_1},\quad \quad} & {{{\alpha \prime}_2} \rightarrow {{\alpha \prime}_2} + 2{{\alpha \prime}_1},\;} & {{{\alpha \prime}_3} \rightarrow {{\alpha \prime}_3},\quad \quad} \\ {{{r\prime}_2}:} & {{{\alpha \prime}_1} \rightarrow {{\alpha \prime}_1} + 2{{\alpha \prime}_2},} & {{{\alpha \prime}_2} \rightarrow - {{\alpha \prime}_2},\quad \quad} & {{{\alpha \prime}_3} \rightarrow {{\alpha \prime}_3} + {{\alpha \prime}_2},} \\ {{{r\prime}_3}:} & {{{\alpha \prime}_1} \rightarrow {{\alpha \prime}_1},\quad \quad} & {{{\alpha \prime}_2} \rightarrow {{\alpha \prime}_2} + {{\alpha \prime}_3},\quad} & {{{\alpha \prime}_3} \rightarrow - {{\alpha \prime}_3},\quad} \\ \end{array}$$
    while the Weyl group \(\mathfrak{W}[A_2^{(2) +}]\) acts as
    $$\begin{array}{*{20}c} {r\prime {\prime _1}:} & {\alpha \prime {\prime _1} \rightarrow - \alpha \prime {\prime _1},\;\;\quad} & {\alpha \prime {\prime _2} \rightarrow \alpha \prime {\prime _2} + 4\alpha \prime {\prime _1},\;} & {\alpha \prime {\prime _3} \rightarrow \alpha \prime {\prime _3},\;\;\quad \quad \;} \\ {r\prime {\prime _2}:} & {\alpha \prime {\prime _1} \rightarrow \alpha \prime {\prime _1} + \alpha \prime {\prime _2},} & {\alpha \prime {\prime _2} \rightarrow - \alpha \prime {\prime _2},\;\;\quad \quad} & {\alpha \prime {\prime _3} \rightarrow \alpha \prime {\prime _3} + \alpha \prime {\prime _2},\;} \\ {r\prime {\prime _3}:} & {\alpha \prime {\prime _1} \rightarrow \alpha \prime {\prime _1},\;\;\quad \;\;} & {\alpha \prime {\prime _2} \rightarrow \alpha \prime {\prime _2} + \alpha \prime {\prime _3},\;\;} & {\alpha \prime {\prime _3} \rightarrow - \alpha \prime {\prime _3}.\;\;\;\;\;\;\;} \\ \end{array}$$
    We see that the reflections coincide, \({\sigma _1} = r_1\prime = r_1{\prime \prime},\,{\sigma _2} = r_2\prime = r_2{\prime \prime},\,{\sigma _3} = r_3\prime = r_3{\prime \prime}\), as well as the scalar products, provided that we set \(2\alpha _1{\prime \prime} = \alpha _1\prime, \,\alpha _2{\prime \prime} = \alpha _2\prime, \,\alpha _3\prime = {\alpha _3}\) and \(\alpha _i\prime = \sqrt 2 {\alpha _i}\). The Coxeter group \(\mathfrak{B}\) generated by the reflections thus preserves the lattices
    $$Q\prime = \sum\limits_i {{\mathbb Z}\alpha \prime _i}\qquad {\rm{and}}\qquad Q\prime \prime = \sum\limits_i {{\mathbb Z}\alpha \prime \prime _i},$$
    (4.61)
    showing explicitly that, in the present case, the lattices preserved by a Coxeter group are not unique — and might not even be dual to each other.
    It follows, of course, that the Weyl groups of the Kac-Moody algebras \(A_1^{+ +}\) and \(A_1^{(2) +}\) are the same,
    $$\mathfrak{W}[A_1^{+ +}] =\mathfrak{W} [A_2^{(2) +}] =\mathfrak{B} .$$
    (4.62)
  • Consider now the Cartan matrix
    $$A\prime \prime \prime = \left({\begin{array}{*{20}c} 2 & {- 6} & 0 \\ {- 1} & 2 & {- 1} \\ 0 & {- 1} & 2 \\\end{array}} \right),$$
    and its symmetrization
    $$S\prime \prime \prime = \left({\begin{array}{*{20}c} {{1 \over 3}} & {- 1} & 0 \\ {- 1} & 2 & {- 1} \\ 0 & {- 1} & 2 \\ \end{array}} \right),$$
    The Weyl group \(\mathfrak{W}[A\prime \prime \prime ]\) of the corresponding Kac-Moody algebra is isomorphic to the Coxeter group \(\mathfrak{B}\) above since, according to the rules, the Coxeter exponents are identical. But the action is now
    $$\begin{array}{*{20}c} {r\prime \prime {\prime _1}}: & {\alpha \prime \prime {\prime _1} \rightarrow - \alpha \prime \prime {\prime _1},\;\;\quad \;} & {\alpha \prime \prime {\prime _2} \rightarrow \alpha \prime \prime {\prime _2} + 6\alpha \prime \prime {\prime _1},} & {\alpha \prime \prime {\prime _3} \rightarrow \alpha \prime \prime {\prime _3}\quad \quad \quad} \\ {r\prime \prime {\prime _2}}: & {\alpha \prime \prime {\prime _1} \rightarrow \alpha \prime \prime {\prime _1} + \alpha \prime \prime {\prime _2},} & {\alpha \prime \prime {\prime _2} \rightarrow - \alpha \prime \prime {\prime _2}\quad \;\;\quad} & {\alpha \prime \prime {\prime _3} \rightarrow \alpha \prime \prime {\prime _3} + \alpha \prime \prime {\prime _2}\;} \\ {r\prime \prime {\prime _3}}:& {\alpha \prime \prime {\prime _1} \rightarrow \alpha \prime \prime {\prime _1},\;\quad \quad} & {\alpha \prime \prime {\prime _2} \rightarrow \alpha \prime \prime {\prime _2} + \alpha \prime \prime {\prime _3},} & {\alpha \prime \prime {\prime _3} \rightarrow - \alpha \prime \prime {\prime _3}\quad \quad} \\ \end{array}$$
    and cannot be made to coincide with the previous action by rescalings of the α i ″’s. One can easily convince oneself of the inequivalence by computing the eigenvalues of the matrices S′, S″ and S′″ with respect to B.

4.8 Hyperbolic Kac-Moody algebras

Hyperbolic Kac-Moody algebras are by definition Lorentzian Kac-Moody algebras with the property that removing any node from their Dynkin diagram leaves one with a Dynkin diagram of affine or finite type. The Weyl group of hyperbolic Kac-Moody algebras is a crystallographic hyperbolic Coxeter group (as defined in Section 3.5). Conversely, any crystallographic hyperbolic Coxeter group is the Weyl group of at least one hyperbolic Kac-Moody algebra. Indeed, consider one of the lattices preserved by the Coxeter group as constructed in Section 3.6. The matrix with entries equal to the d ij of that section is the Cartan matrix of a Kac-Moody algebra that has this given Coxeter group as Weyl group.

The hyperbolic Kac-Moody algebras have been classified in [154] and exist only up to rank 10 (see also [59]). In rank 10, there are four possibilities, known as \({E_{10}} \equiv E_8^{+ +},\,B{E_{10}} \equiv B_8^{+ +},\,D{E_{10}} \equiv D_8^{+ +}\) and \(C{E_{10}} \equiv A_{15}^{(2) +}\), BE10 and CE10 being dual to each other and possessing the same Weyl group (the notation will be explained below).

4.8.1 The fundamental domain \({\mathcal F}\)

For a hyperbolic Kac-Moody algebra, the fundamental weights Λ i are timelike or null and lie within the (say) past lightcone. Similarly, the fundamental Weyl chamber \({\mathcal F}\) defined by \(\{v \in \mathcal{F} \Leftrightarrow (v \vert \alpha_i) \geq 0 \}\) also lies within the past lightcone and is a fundamental region for the action of the Weyl group on the Tits cone, which coincides in fact with the past light cone. All these properties carries over from our discussion of hyperbolic Coxeter groups in Section 3.

The positive imaginary roots α K of the algebra fulfill (α K j ) ≥ 0 (with, for any K, strict inequality for at least one i) and hence, since they are non-spacelike, must lie in the future light cone. Recall indeed that the scalar product of two non-spacelike vectors with the same time orientation is non-positive. For this reason, it is also of interest to consider the action of the Weyl group on the future lightcone, obtained from the action on the past lightcone by mere changes of signs. A fundamental region is clearly given by \(- \mathcal{F}\). Any imaginary root is Weyl-conjugated to one that lies in \(- \mathcal{F}\).

4.8.2 Roots and the root lattice

We have mentioned that not all points on the root lattice Q of a Kac-Moody algebras are actually roots. For hyperbolic algebras, there exists a simple criterion which enables one to determine whether a point on the root lattice is a root or not. We give it first in the case where all simple roots have equal length squared (assumed equal to two).

Theorem: Consider a hyperbolic Kac-Moody algebra such that (α i |α i ) = 2 for all simple roots α i . Then, any point α on the root lattice Q with (α|α) ≤ 2 is a root (note that (α|α) is even). In particular, the set of real roots is the set of points on the root lattice with (α|α) = 2, while the set of imaginary roots is the set of points on the root lattice (minus the origin) with (α|α) ≤ 0. For a proof, see [116], Chapter 5.

The version of this theorem applicable to Kac-Moody algebras with different simple root lengths is the following.

Theorem: Consider a hyperbolic algebra with root lattice Q. Let a be the smallest length squared of the simple roots, a = min i (α i |α i ). Then we have:
  • The set of all short real roots is {αQ| (α|α) = a}.

  • The set of all real roots is
    $$\left\{{\alpha = \sum\limits_i {{k_i}} {\alpha _i} \in Q\,\vert \,(\alpha \vert \alpha) > 0\;{\rm{and}}\;{k_i}{{({\alpha _i}\vert {\alpha _i})} \over {(\alpha \vert \alpha)}} \in \;{\mathbb Z}\forall i} \right\}.$$
  • The set of all imaginary roots is the set of points on the root lattice (minus the origin) with (α|α) ≤ 0.

For a proof, we refer again to [116], Chapter 5.

We shall illustrate these theorems in the examples below. Note that it follows in particular from the theorems that if α is an imaginary root, all its integer multiples are also imaginary roots.

4.8.3 Examples

We discuss here briefly two examples, namely \(A_1^{+ +}\), for which all simple roots have equal length, and \(A_2^{(2) +}\), with respective Dynkin diagrams shown in Figures 17 and 18.
Figure 17

The Dynkin diagram of the hyperbolic Kac-Moody algebra \(A_1^{+ +}\). This algebra is obtained through a standard overextension of the finite Lie algebra A1.

Figure 18

The Dynkin diagram of the hyperbolic Kac-Moody algebra \(A_2^{(2) +}\). This algebra is obtained through a Lorentzian extension of the twisted affine Kac-Moody algebra \(A_2^{(2)}\).

4.8.3.1 The Kac-Moody Algebra \(A_1^{+ +}\)
This is the algebra associated with vacuum four-dimensional Einstein gravity and the BKL billiard. We encountered its Weyl group PGL(2, ℤ) already in Section 3.1.1. The algebra is also denoted AE3, or H3. The Cartan matrix is
$$\left({\begin{array}{*{20}c} 2 & {- 2} & 0 \\ {- 2} & 2 & {- 1} \\ 0 & {- 1} & 2 \\ \end{array}} \right).$$
(4.63)
As it follows from our analysis in Section 3.1.1, the simple roots may be identified with the following linear forms α i (β) in the three-dimensional space of the β i ’s,
$${\alpha _1}(\beta) = 2{\beta ^1},\qquad {\alpha _2}(\beta) = {\beta ^2} - {\beta ^1},\qquad {\alpha _3}(\beta) = {\beta ^3} - {\beta ^2}$$
(4.64)
with scalar product
$$(F\vert G) = \sum\limits_i {{F_i}} {G_i} - {1 \over 2}\left({\sum\limits_i {{F_i}}} \right)\left({\sum\limits_i {{G_i}}} \right)$$
(4.65)
for two linear forms F = F i β i and G = G i β i . It is sometimes convenient to analyze the root system in terms of an “affine” level that counts the number of times the root α3 occurs: The root 1 + 2 + ℓα3 has by definition level 13. We shall consider here only positive roots for which k, m, ≥ 0.

Applying the first theorem, one easily verifies that the only positive roots at level zero are the roots 1 + 2, |km| ≤ 1 (k, m ≥ 0) of the affine subalgebra \(A_1^ +\). When k = m, the root is imaginary and has length squared equal to zero. When |km| = 1, the root is real and has length squared equal to two.

Similarly, the only roots at level one are (m + a)α1 + 2 + α3 with a2m, i.e., \(- \left[ {\sqrt m} \right] \leq a \leq \left[ {\sqrt m} \right]\). Whenever \(\sqrt m\) is an integer, the roots \(\left({m \pm \sqrt m} \right){\alpha _1} + m{\alpha _2} + {\alpha _3}\) have squared length equal to two and are real. The roots (m + a)α1 + 2 + α3 with a2 < m are imaginary and have squared length equal to 2(a2 + 1 − m) ≤ 0. In particular, the root m(α1 + α2) + α3 has length squared equal to 2(1 − m). Of all the roots at level one with m > 1, these are the only ones that are in the fundamental domain \(-\mathcal{F}\) (i.e., that fulfill (β|αi) ≤ 0). When m = 1, none of the level-1 roots is in \(-\mathcal{F}\) and is either in the Weyl orbit of α1 + α2, or in the Weyl orbit of α3.

We leave it to the reader to verify that the roots at level two that are in the fundamental domain \(-\mathcal{F}\) take the form (m − 1)α1 + 2 + 2α3 and m(α1 + α2) + 2α3 with m ≥ 4. Further information on the roots of \(A_1^{+ +}\) may be found in [116], Chapter 11, page 215.

4.8.3.2 The Kac-Moody Algebra \(A_2^{(2) +}\)
This is the algebra associated with the Einstein-Maxwell theory (see Section 7). The notation will be explained in Section 4.9. The Cartan matrix is
$$\left({\begin{array}{*{20}c} 2 & {- 4} & 0 \\ {- 1} & 2 & {- 1} \\ 0 & {- 1} & 2 \\ \end{array}} \right),$$
(4.66)
and there are now two lengths for the simple roots. The scalar products are
$$({\alpha _1}\vert {\alpha _1}) = {1 \over 2},\qquad ({\alpha _1}\vert {\alpha _2}) = - 1 = ({\alpha _2}\vert {\alpha _1}),\qquad ({\alpha _2}\vert {\alpha _2}) = 2{.}$$
(4.67)
One may realize the simple roots as the linear forms
$${\alpha _1}(\beta) = {\beta ^1},\qquad {\alpha _2}(\beta) = {\beta ^2} - {\beta ^1},\qquad {\alpha _3}(\beta) = {\beta ^3} - {\beta ^2}$$
(4.68)
in the three-dimensional space of the β i ’s with scalar product Equation (4.65).

The real roots, which are Weyl conjugate to one of the simple roots α1 or α2 (α3 is in the same Weyl orbit as α2), divide into long and short real roots. The long real roots are the vectors on the root lattice with squared length equal to two that fulfill the extra condition in the theorem. This condition expresses here that the coefficient of α1 should be a multiple of 4. The short real roots are the vectors on the root lattice with length squared equal to one-half. The imaginary roots are all the vectors on the root lattice with length squared ≤ 0.

We define again the level as counting the number of times the root α3 occurs. The positive roots at level zero are the positive roots of the twisted affine algebra \(A_2^{(2)}\), namely, α1 and (2m + a)α1+2, m = 1, 2, 3, ⋯, with a = −2, −1, 0, 1, 2 for m odd and a = −1, 0, 1 for m odd. Although belonging to the root lattice and of length squared equal to two, the vectors (2m ± 2)α1 + 2 are not long real roots when m is even because they fail to satisfy the condition that the coefficient (2m ± 2) of α1 is a multiple of 4. The roots at level zero are all real, except when a = 0, in which case the roots m(2α1 + α2) have zero norm.

To get the long real roots at level one, we first determine the vectors α = α3 + 1 + 2 of squared length equal to two. The condition (α|α) = 2 easily leads to m = p2 for some integer p ≥ 0 and k = 2p2 ± 2p = 2p(p ± 1). Since k is automatically a multiple of 4 for all p = 0, 1, 2, 3, ⋯, the corresponding vectors are all long real roots. Similarly, the short real roots at level one are found to be (2p2 + 1)α1 + (p2 + p + 1)α2 + α3 and (2p2 + 4p + 3)α1 + (p2 + p + 1)α2 + α3 for p a non-negative integer.

Finally, the imaginary roots at level one in the fundamental domain \(-\mathcal{F}\) read (2m − 1)α1 + 2 + α3 and 21 + 2 + α3 where m is an integer greater than or equal to 2. The first roots have length squared equal to \(- 2m + {5 \over 2}\), the second have length squared equal to −2m + 2.

4.9 Overextensions of finite-dimensional Lie algebras

An interesting class of Lorentzian Kac-Moody algebras can be constructed by adding simple roots to finite-dimensional simple Lie algebras in a particular way which will be described below. These are called “overextensions”.

In this section, we let \(\mathfrak{g}\) be a complex, finite-dimensional, simple Lie algebra of rank r, with simple roots α1, ⋯, α r . As stated above, normalize the roots so that the long roots have length squared equal to 2 (the short roots, if any, have then length squared equal to 1 (or 2/3 for G2)). The roots of simply-laced algebras are regarded as long roots.

Let α = ∑ i n i α i , n i ≥ 0 be a positive root. One defines the height of α as
$${\rm{ht}}(\alpha) = \sum\limits_i {{n_i}} .$$
(4.69)
Among the roots of \(\mathfrak{g}\), there is a unique one that has highest height, called the highest root. We denote it by θ. It is long and it fulfills the property that (θ|α i ) ≥ 0 for all simple roots α i , and
$$2{{({\alpha _i}\vert \theta)} \over {(\theta \vert \theta)}} \in\mathbb{Z} ,\qquad 2{{(\theta \vert {\alpha _i})} \over {({\alpha _i}\vert {\alpha _i})}} \in \mathbb{Z}$$
(4.70)
(see, e.g., [85]). We denote by V the r-dimensional Euclidean vector space spanned by α i (i = 1, ⋯, r). Let M2 be the two-dimensional Minkowski space with basis vectors u and v so that (u|u) = (v|v) = 0 and (u|v) = 1. The metric in the space VM2 has clearly Minkowskian signature (−, +, +, ⋯, +) so that any Kac-Moody algebra whose simple roots span VM2 is necessarily Lorentzian.

4.9.1 Untwisted overextensions

The standard overextensions \(\mathfrak{g}^{++}\) are obtained by adding to the original roots of \(\mathfrak{g}\) the roots \(\alpha _i\prime\) The matrix \({A_{ij}} = 2{{\left({{\alpha _i}\vert {\alpha _j}} \right)} \over {\left({{\alpha _i}\vert {\alpha _i}} \right)}}\) where i, j = −1, 0, 1, ⋯, r is a (generalized) Cartan matrix and defines indeed a Kac-Moody algebra.

The root α0 is called the affine root and the algebra \(\mathfrak{g}^+\) (\(\mathfrak{g}^{(1)}\) in Kac’s notations [116]) with roots α0, α1, ⋯, α r is the untwisted affine extension of \(\mathfrak{g}\). The root α−1 is known as the overextended root. One clearly has \({\rm{rank}}({\mathfrak{g}^{+ +}}) = {\rm{rank}}(\mathfrak{g}) + 2\). The overextended root has vanishing scalar product with all other simple roots except α0. One has explicitly (α−1 |α−1) = 2 = (α0|α0) and (α−1 |α0) = −1, which shows that the overextended root is attached to the affine root (and only to the affine root) with a single link.

Of these Lorentzian algebras, the following ones are hyperbolic:
  • \(A_k^{+ +}(k \leq 7)\),

  • \(B_k^{+ +}(k \leq 8)\),

  • \(C_k^{+ +}(k \leq 4)\),

  • \(D_k^{+ +}(k \leq 8)\),

  • \(G_2^{+ +}\),

  • \(F_4^{+ +}\),

  • \(E_k^{+ +}(k = 6,\,7,\,8)\).

The algebras \(B_8^{+ +},\,D_8^{+ +}\) and \(E_8^{+ +}\) are also denoted BE10, DE10 and E10, respectively.
4.9.1.1 A special property of E10

Of these maximal rank hyperbolic algebras, E10 plays a very special role. Indeed, one can verify that the determinant of its Cartan matrix is equal to −1. It follows that the lattice of E10 is self-dual, i.e., that the fundamental weights belong to the root lattice of E10. In view of the above theorem on roots of hyperbolic algebras and of the hyperboliticity of E10, the fundamental weights of E10 are actually (imaginary) roots since they are non-spacelike. The root lattice of E10 is the only Lorentzian, even, self-dual lattice in 10 dimensions (these lattices exist only in 2 mod 8 dimensions).

4.9.2 Root systems in Euclidean space

In order to describe the “twisted” overextensions, we need to introduce the concept of a “root system”.

A root system in a real Euclidean space V is by definition a finite subset Δ of nonzero elements of V obeying the following two conditions:
$$\Delta\;\;{\rm{spans}}\;\;V,$$
(4.71)
$$\forall \alpha ,\,\beta \in \Delta :\qquad \left\{{\begin{array}{*{20}c} {{A_{\alpha ,\beta}} = 2{{(\alpha \vert \beta)} \over {(\beta \vert \beta)}} \in {\mathbb Z},} \\ {\beta - {A_{\beta ,\alpha}}\,\alpha \in \Delta .\quad \quad \;} \\ \end{array}} \right.$$
(4.72)
The elements of Δ are called the roots. From the definition one can prove the following properties [93]:
  1. 1.

    If α ∈ Δ, then −α ∈ Δ.

     
  2. 2.

    If α ∈ Δ, then the only elements of Δ proportional to α are ±½α, ±α, ±2α. If only ±α occurs (for all roots α), the root system is said to be reduced (proper in “Araki terminology” [5]).

     
  3. 3.

    If α, β ∈ Δ, then 0 ≤ A α,β A β,α ≤ 4, i.e., A α,β = 0, ±1, ±2, ±3, ±4; the last occurrence appearing only for β = ±2α, i.e., for nonreduced systems. (The proof of this point requires the use of the Schwarz inequality.)

     
  4. 4.

    If α, β ∈ Δ are not proportional to each other and (α|α) ≤ (β|β) then A α,β = 0, ±1. Moreover if (α|β) = 0, then (β|β) = (α|α), 2 (α|α), or 3 (α|α).

     
  5. 5.

    If If α, β ∈ Δ, but αβ ∉ Δ ∪ 0, then (α|β) ≤ 0 and, as a consequence, if α, β ∈ Δ but α ± β ∉ Δ ∪ 0 then (α|β) = 0. That (α|β) ≤ 0 can be seen as follows. Clearly, α and β can be assumed to be linearly independent14. Now, assume (α|β) > 0. By the previous point, A α,β = 1 or A β,α = 1. But then either αA α,β β = αβ ∈ Δ or −(βA β,α α) = αβ ∈ Δ by (4.72), contrary to the assumption. This proves that (α|β) ≤ 0.

     

Since Δ spans the vector space V, one can chose a basis {α i } of elements of V within Δ. This can furthermore be achieved in such a way the α i enjoy the standard properties of simple roots of Lie algebras so that in particular the concepts of positive, negative and highest roots can be introduced [93].

All the abstract root systems in Euclidean space have been classified (see, e.g., [93]) with the following results:
  • The most general root system is obtained by taking a union of irreducible root systems. An irreducible root system is one that cannot be decomposed into two disjoint nonempty orthogonal subsets.

  • The irreducible reduced root systems are simply the root systems of finite-dimensional simple Lie algebras (A n with n ≥ 1, B n with n ≥ 3, C n with n ≥ 2, D n with n ≥ 4, G2, F4, E6, E7 and E8).

  • Irreducible nonreduced root systems are all given by the so-called (BC) n -systems. A (BC) n -system is obtained by combining the root system of the algebra B n with the root system of the algebra C n in such a way that the long roots of B n are the short roots of C n . There are in that case three different root lengths. Explicitly Δ is given by the n unit vectors \({\vec e_k}\) and their opposite \(- {\vec e_k}\) along the Cartesian axis of an n-dimensional Euclidean space, the 2n vectors \(\pm 2{\vec e_k}\) obtained by multiplying the previous vectors by 2 and the 2n(n − 1) diagonal vectors \(\pm {\vec e_k} \pm {\vec e_{k\prime}}\), with kk′ and k, k′ = 1, …, n. The n = 3 case is pictured in Figure 19. The Dynkin diagram of (BC) r is the Dynkin diagram of B r with a double circle ⊚ over the simple short root, say α1, to indicate that 2α1 is also a root.

Figure 19

The nonreduced (BC)2- and (BC)3-root systems. In each case, the highest root θ is displayed.

It is sometimes convenient to rescale the roots by the factor \((1/\sqrt 2)\) so that the highest root θ = 2(α1 + α2 + ⋯ + α r ) [93] of the (BC)-system has length 2 instead of 4.

4.9.3 Twisted overextensions

We follow closely [95]. Twisted affine algebras are related to either the (BC)-root systems or to extensions by the highest short root (see [116], Proposition 6.4).

4.9.3.1 Twisted overextensions associated with the (BC)-root systems

These are the overextensions relevant for some of the gravitational billiards. The construction proceeds as for the untwisted overextensions, but the starting point is now the (BC) r root system with rescaled roots. The highest root has length squared equal to 2 and has non-vanishing scalar product only with α r ((α r |θ) = 1). The overextension procedure (defined by the same formulas as in the untwisted case) yields the algebra \((BC)_r^{+ +}\), also denoted \(A_{2r}^{(2) +}\).

There is an alternative overextension \(A_{2r}^{(2)\prime +}\) that can be defined by starting this time with the algebra C r but taking one-half the highest root of C r to make the extension (see [116], formula in Paragraph 6.4, bottom of page 84). The formulas for α0 and α−1 are 2α0 = uθ and 2α−1 = −uv (where θ is now the highest root of C r ). The Dynkin diagram of \(A_{2r}^{(2)\prime +}\) is dual to that of \(A_{2r}^{(2) +}\). (Duality amounts to reversing the arrows in the Dynkin diagram, i.e., replacing the (generalized) Cartan matrix by its transpose.)

The algebras \(A_{2r}^{(2) +}\) and \(A_{2r}^{(2)\prime +}\) have rank r + 2 and are hyperbolic for r ≤ 4. The intermediate affine algebras are in all cases the twisted affine algebras \(A_{2r}^{(2)}\). We shall see in Section 7 that by coupling to three-dimensional gravity a coset model \({\mathcal G}/{\mathcal K}({\mathcal G})\), where the so-called restricted root system (see Section 6) of the (real) Lie algebra \(\mathfrak{g}\) of the Lie group \({\mathcal G}\) is of (BC) r -type, one can realize all the \(A_{2r}^{(2) +}\) algebras.

4.9.3.2 Twisted overextensions associated with the highest short root
We denote by θ s the unique short root of heighest weight. It exists only for non-simply laced algebras and has length 1 (or 2/3 for G2). The twisted overextensions are defined as the standard overextensions but one uses instead the highest short root θ s . The formulas for the affine and overextended roots are
$${\alpha _0} = u - {\theta _s},\qquad {\alpha _{- 1}} = - u - {1 \over 2}v,\qquad ({\mathfrak g} = {B_r},{C_r},{F_4})$$
or
$${\alpha _0} = u - {\theta _s},\qquad {\alpha _{- 1}} = - u - {1 \over 3}v,\qquad ({\mathfrak g} = {G_2}).$$
(We choose the overextended root to have the same length as the affine root and to be attached to it with a single link. This choice is motivated by considerations of simplicity and yields the fourth rank ten hyperbolic algebra when \(\mathfrak{g}={C_8}\).)
The affine extensions generated by α0, ⋯, α r are respectively the twisted affine algebras \(D_{r + 1}^{(2)}\,(\mathfrak{g} = {B_r}),\,A_{2r - 1}^{(2)}\,(\mathfrak{g} = {C_r}),\,E_6^{(2)}\,(\mathfrak{g} = {F_4})\) and \(D_4^{(3)}\,(\mathfrak{g} = {G_2})\). These twisted affine algebras are related to external automorphisms of Dr+1, A2r−1, E6 and D4, respectively (the same holds for \(A_{2r}^{(2)}\) above) [116]. The corresponding twisted overextensions have the following features.
  • The overextensions \(D_{r + 1}^{(2) +}\) have rank r + 2 and are hyperbolic for r ≤ 4.

  • The overextensions \(A_{2r - 1}^{(2) +}\) have rank r + 2 and are hyperbolic for r ≤ 8. The last hyperbolic case, r = 8, yields the algebra \(A_{15}^{(2) +}\), also denoted CE10. It is the fourth rank-10 hyperbolic algebra, besides E10, BE10 and DE10.

  • The overextensions \(E_6^{(2) +}\) (rank 6) and \(D_4^{(3) +}\) (rank 4) are hyperbolic.

We list in Table 14 the Dynkin diagrams of all twisted overextensions.
Table 14

Twisted overextended Kac-Moody algebras.

Name

Dynkin diagram

\(A_2^{(2) +}\)

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\(A_2^{(2)\prime}\)

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\(A_{2n}^{(2) +}\,(n \geq 2)\)

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\(A_{2n}^{(2)\prime +}\,(n \geq 2)\)

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\(A_{2n - 1}^{(2) +}\,(n \geq 3)\)

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\(D_{n + 1}^{(2) +}\)

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\(E_6^{(2) +}\)

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\(D_4^{(3) +}\)

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A satisfactory feature of the class of overextensions (standard and twisted) is that it is closed under duality. For instance, \(A_{2r - 1}^{(2) +}\) is dual to \(B_r^{+ +}\). In fact, one could get the twisted overextensions associated with the highest short root from the standard overextensions precisely by requiring closure under duality. A similar feature already holds for the affine algebras.

Note also that while not all hyperbolic Kac-Moody algebras are symmetrizable, the ones that are obtained through the process of overextension are.

4.9.4 Algebras of Gaberdiel-Olive-West type

One can further extend the overextended algebras to get “triple extensions” or “very extended algebras”. This is done by adding a further simple root attached with a single link to the overextended root of Section 4.9. For instance, in the case of E10, one gets E11 with the Dynkin diagram displayed in Figure 20. These algebras are Lorentzian, but not hyperbolic.
Figure 20

The Dynkin diagram of E11. Labels 0, −1 and −2 enumerate the nodes corresponding, respectively, to the affine root α0, the overextended root α−1 and the “very extended” root α−2.

The very extended algebras belong to a more general class of algebras considered by Gaberdiel, Olive and West in [86]. These are defined to be algebras with a connected Dynkin diagram that possesses at least one node whose deletion yields a diagram with connected components that are of finite type except for at most one of affine type. For a hyperbolic algebra, the deletion of any node should fulfill this condition. The algebras of Gaberdiel, Olive and West are Lorentzian if not of finite or affine type [153, 86]. They include the overextensions of Section 4.9. The untwisted and twisted very extended algebras are clearly also of this type, since removing the affine root gives a diagram with the requested properties.

Higher order extensions with special additional properties have been investigated in [78].

4.10 Regular subalgebras of Kac-Moody algebras

This section is based on [96].

4.10.1 Definitions

Let \({\mathfrak g}\) be a Kac-Moody algebra, and let \({\bar {\mathfrak g}}\) be a subalgebra of \({\mathfrak g}\) with triangular decomposition \({\bar {\mathfrak g}} = {{\bar {\mathfrak n}}_ -} \oplus {\bar {\mathfrak h}} \oplus {{\bar {\mathfrak n}}_ +}\). We assume that \({\bar {\mathfrak g}}\) is canonically embedded in \({\mathfrak g}\), i.e., that the Cartan subalgebra \({\bar {\mathfrak h}}\) of \({\bar {\mathfrak g}}\) is a subalgebra of the Cartan subalgebra \({\mathfrak h}\) of \({\mathfrak g}\), \({\bar {\mathfrak h}} \subset {\mathfrak h}\), so that \({\bar {\mathfrak h}} = {\bar {\mathfrak g}} \cap {\mathfrak h}\). We shall say that \({\bar {\mathfrak g}}\) is regularly embedded in \({\mathfrak g}\) (and call it a “regular subalgebra”) if and only if two conditions are fulfilled: (i) The root generators of \({\bar {\mathfrak g}}\) are root generators of \({\mathfrak g}\), and (ii) the simple roots of \({\bar {\mathfrak g}}\) are real roots of \({\mathfrak g}\). It follows that the Weyl group of \({\bar {\mathfrak g}}\) is a subgroup of the Weyl group of \({\mathfrak g}\) and that the root lattice of \({\bar {\mathfrak g}}\) is a sublattice of the root lattice of \({\mathfrak g}\).

The second condition is automatic in the finite-dimensional case where there are only real roots. It must be separately imposed in the general case. Consider for instance the rank 2 Kac-Moody algebra \({\mathfrak g}\) with Cartan matrix
$$\left({\begin{array}{*{20}c} 2 & {- 3} \\ {- 3} & 2 \\ \end{array}} \right).$$
Let
$$x = {1 \over {\sqrt 3}}[{e_1},{e_2}],$$
(4.73)
$$y = {1 \over {\sqrt 3}}[{f_1},{f_2}],$$
(4.74)
$$z = - ({h_1} + {h_2}).$$
(4.75)
It is easy to verify that x, y, z define an A1 subalgebra of \({\mathfrak g}\) since [z, x] = 2x, [z, y] = −2y and [x, y] = z. Moreover, the Cartan subalgebra of A1 is a subalgebra of the Cartan subalgebra of g, and the step operators of A1 are step operators of \({\mathfrak g}\). However, the simple root α = α1 + α2 of A1 (which is an A1-real root since A1 is finite-dimensional), is an imaginary root of \({\mathfrak {g}}: {\alpha _1} + {\alpha _2}\) has norm squared equal to −2. Even though the root lattice of A1 (namely, {±α}) is a sublattice of the root lattice of \({\mathfrak g}\), the reflection in α is not a Weyl reflection of \({\mathfrak g}\). According to our definition, this embedding of A1 in \({\mathfrak g}\) is not a regular embedding.

4.10.2 Examples — Regular subalgebras of E10

We shall describe some regular subalgebras of E10. The Dynkin diagram of E10 is displayed in Figure 21.
Figure 21

The Dynkin diagram of E10. Labels 1, ⋯, 7 and 10 enumerate the nodes corresponding the regular E8 subalgebra discussed in the text.

4.10.2.1 \({A_9} \subset {\mathcal B} \subset{E_{10}}\)
A first, simple, example of a regular embedding is the embedding of A9 in E10 which will be used to define the level when trying to reformulate eleven-dimensional supergravity as a nonlinear sigma model. This is not a maximal embedding since one can find a proper subalgebra \({\mathcal B}\) of E10 that contains A9. One may take for \({\mathcal B}\) the Kac-Moody subalgebra of E10 generated by the operators at levels 0 and ±2, which is a subalgebra of the algebra containing all operators of even level15. It is regularly embedded in E10. Its Dynkin diagram is shown in Figure 22.
Figure 22

The Dynkin diagram of \({\mathcal B} \equiv E_7^{+ + +}\). The root without number is the root denoted 10 in the text.

In terms of the simple roots of E10, the simple roots of \({\mathcal B}\) are α1 through α9 and 10 = 2α10 + α1 + 2α2 + 3α3 + 2α4 + α5. The algebra \({\mathcal B}\) is Lorentzian but not hyperbolic. It can be identified with the “very extended” algebra \(E_7^{+ + +}\) [86].

4.10.2.2 DE10E10

In [67], Dynkin has given a method for finding all maximal regular subalgebras of finite-dimensional simple Lie algebras. The method is based on using the highest root and is not generalizable as such to general Kac-Moody algebras for which there is no highest root. Nevertherless, it is useful for constructing regular embeddings of overextensions of finite-dimensional simple Lie algebras. We illustrate this point in the case of E8 and its overextension \({E_{10}} \equiv E_8^{+ +}\). In the notation of Figure 21, the simple roots of E8 (which is regularly embedded in E10) are α1, ⋯, α7 and α10.

Applying Dynkin’s procedure to E8, one easily finds that D8 can be regularly embedded in E8. The simple roots of D8E8 are α2, α3, α4, α5, α6, α7, α10 and \(\beta \equiv - {\theta _{{E_8}}}\), where
$${\theta _{{E_8}}} = 3{\alpha _{10}} + 6{\alpha _3} + 4{\alpha _2} + 2{\alpha _1} + 5{\alpha _4} + 4{\alpha _5} + 3{\alpha _6} + 2{\alpha _7}$$
(4.76)
is the highest root of E8. One can replace this embedding, in which a simple root of D8, namely β, is a negative root of E8 (and the corresponding raising operator of D8 is a lowering operator for E8), by an equivalent one in which all simple roots of D8 are positive roots of E8.
This is done as follows. It is reasonable to guess that the searched-for Weyl element that maps the “old” D8 on the “new” D8 is some product of the Weyl reflections in the four E8-roots orthogonal to the simple roots α3, α4, α5, α6 and α7, expected to be shared (as simple roots) by E8, the old D8 and the new D8 — and therefore to be invariant under the searched-for Weyl element. This guess turns out to be correct: Under the action of the product of the commuting E8-Weyl reflections in the E8-roots μ1 = 2α1 + 3α2 + 5α3 + 4α4 + 3α5 + 2α6 + α7 + 3α10 and μ2 = 2α1+4α2 +5α3+4α4+3α5 + 2α6+α7+2α10, the set of D8-roots {α2, α3, α4, α5, α6, α7, α10, β} is mapped on the equivalent set of positive roots {α10, α3, α4, α5, α6, α7, α2, β}, where
$$\bar \beta = 2{\alpha _1} + 3{\alpha _2} + 4{\alpha _3} + 3{\alpha _4} + 2{\alpha _5} + {\alpha _6} + 2{\alpha _{10}}.$$
(4.77)
In this equivalent embedding, all raising operators of D8 are also raising operators of E8. What is more, the highest root of D8,
$${\theta _{{D_8}}} = {\alpha _{10}} + 2{\alpha _3} + 2{\alpha _4} + 2{\alpha _5} + 2{\alpha _6} + 2{\alpha _7} + {\alpha _2} + \bar \beta$$
(4.78)
is equal to the highest root of E8. Because of this, the affine root α8 of the untwisted affine extension \(E_8^{+}\) can be identified with the affine root of \(D_8^{+}\), and the overextended root α9 can also be taken to be the same. Hence, DE10 can be regularly embedded in E10 (see Figure 23).
Figure 23

\(D{E_{10}} \equiv D_8^{+ +}\) regularly embedded in E10. Labels 2, ⋯, 10 represent the simple roots α2, ⋯, α10 of E10 and the unlabeled node corresponds to the positive root \(\bar \beta = 2{\alpha _1} + 3{\alpha _2} + 4{\alpha _3} + 3{\alpha _4} + 2{\alpha _5} + {\alpha _6} + 2{\alpha _{10}}\).

The embedding just described is in fact relevant to string theory and has been discussed from various points of view in previous papers [125, 23]. By dimensional reduction of the bosonic sector of eleven-dimensional supergravity on a circle, one gets, after dropping the Kaluza-Klein vector and the 3-form, the bosonic sector of pure \({\mathcal N} = 1\) ten-dimensional supergravity. The simple roots of DE10 are the symmetry walls and the electric and magnetic walls of the 2-form and coincide with the positive roots given above [45]. A similar construction shows that \(A_8^{+ +}\) can be regularly embedded in E10, and that DE10 can be regularly embedded in \(B_8^{+ +}\). See [106] for a recent discussion of DE10 in the context of Type I supergravity.

4.10.3 Further properties

As we have just seen, the raising operators of \({\bar {\mathfrak g}}\) might be raising or lowering operators of \({\mathfrak g}\). We shall consider here only the case when the positive (respectively, negative) root generators of \({\bar {\mathfrak g}}\) are also positive (respectively, negative) root generators of \({\mathfrak g}\), so that \({{\bar {\mathfrak n}}_ -} = {{\mathfrak n}_ -} \cap {\bar {\mathfrak g}}\) and \({{\bar {\mathfrak n}}_ +} = {{\mathfrak n}_ +} \cap {\bar {\mathfrak g}}\) (“positive regular embeddings”). This will always be assumed from now on.

In the finite-dimensional case, there is a useful criterion to determine regular algebras from subsets of roots. This criterion, which does not use the highest root, has been generalized to Kac-Moody algebras in [76]. It covers also non-maximal regular subalgebras and goes as follows:

Theorem: Let \(\Phi _{{\rm{real}}}^ +\) be the set of positive real roots of a Kac-Moody algebra \({\mathfrak g}\). Let \({\gamma _1},\, \ldots, \,{\gamma _n} \in \Phi _{{\rm{real}}}^ +\) be chosen such that none of the differences γ i γ j is a root of \({\mathfrak g}\). Assume furthermore that the γ i ’s are such that the matrix C = [C ij ] = [2 (γ i |γ j ) / (γ i |γ i )] has non-vanishing determinant. For each 1 ≤ in, choose non-zero root vectors E i and F i in the one-dimensional root spaces corresponding to the positive real roots γ i and the negative real roots −γ i , respectively, and let H i = [E i , F i ] be the corresponding element in the Cartan subalgebra of \({\mathfrak g}\). Then, the (regular) subalgebra of \({\mathfrak g}\) generated by {E i , F i , H i }, i = 1, ⋯, n, is a Kac-Moody algebra with Cartan matrix [C ij ].

Proof: The proof of this theorem is given in [76]. Note that the Cartan integers \(2{{({\gamma _i}\vert {\gamma _j})} \over {({\gamma _i}\vert {\gamma _i})}}\) are indeed integers (because the γ i ’s are positive real roots), which are non-positive (because γ i γ j is not a root), so that [C ij ] is a Cartan matrix.

4.10.3.1 Comments
  1. 1.

    When the Cartan matrix is degenerate, the corresponding Kac-Moody algebra has nontrivial ideals [116]. Verifying that the Chevalley-Serre relations are fulfilled is not sufficient to guarantee that one gets the Kac-Moody algebra corresponding to the Cartan matrix [C ij ] since there might be non-trivial quotients. Situations in which the algebra generated by the set {E i , F i , H i } is the quotient of the Kac-Moody algebra with Cartan matrix [C ij ] by a non-trivial ideal were discussed in [96].

     
  2. 2.

    If the matrix [C ij ] is decomposable, say C = DE with D and E indecomposable, then the Kac-Moody algebra \({\mathbb K}{\mathbb M}(C)\) generated by C is the direct sum of the Kac-Moody algebra \({\mathbb K}{\mathbb M}(D)\) generated by D and the Kac-Moody algebra \({\mathbb K}{\mathbb M}(E)\) generated by E. The subalgebras \({\mathbb K}{\mathbb M}(D)\) and \({\mathbb K}{\mathbb M}(E)\) are ideals. If C has non-vanishing determinant, then both D and E have non-vanishing determinant. Accordingly, \({\mathbb K}{\mathbb M}(D)\) and \({\mathbb K}{\mathbb M}(E)\) are simple [116] and hence, either occur faithfully or trivially. Because the generators E i are linearly independent, both \({\mathbb K}{\mathbb M}(D)\) and \({\mathbb K}{\mathbb M}(E)\) occur faithfully. Therefore, in the above theorem the only case that requires special treatment is when the Cartan matrix C has vanishing determinant.

     
As we have mentioned above, it is convenient to universally normalize the Killing form of Kac-Moody algebras in such a way that the long real roots have always the same squared length, conveniently taken equal to two. It is then easily seen that the Killing form of any regular Kac-Moody subalgebra of E10 coincides with the invariant form induced from the Killing form of E10 through the embedding since E10 is “simply laced”. This property does not hold for non-regular embeddings as the example given in Section 4.1 shows: The subalgebra A1 considered there has an induced form equal to minus the standard Killing form.

5 Kac-Moody Billiards I — The Case of Split Real Forms

In this section we will begin to explore in more detail the correspondence between Lorentzian Coxeter groups and the limiting behavior of the dynamics of gravitational theories close to a spacelike singularity.

We have seen in Section 2 that in the BKL-limit, the dynamics of gravitational theories is equivalent to a billiard dynamics in a region of hyperbolic space. In the generic case, the billiard region has no particular feature. However, we have seen in Section 3 that in the case of pure gravity in four spacetime dimensions, the billiard region has the remarkable property of being the fundamental domain of the Coxeter group PGL(2, ℤ) acting on two-dimensional hyperbolic space.

This is not an accident. Indeed, this feature arises for all gravitational theories whose toroidal dimensional reduction to three dimensions exhibits hidden symmetries, in the sense that the reduced theory can be reformulated as three-dimensional gravity coupled to a nonlinear sigma-model based on \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\), where \({\mathcal K}({{\mathcal U}_3})\) is the maximal compact subgroup of \({{\mathcal U}_3}\). The “hidden” symmetry group \({{\mathcal U}_3}\) is also called, by a generalization of language, “the U-duality group” [142]. This situation covers the cases of pure gravity in any spacetime dimension, as well as all known super-gravity models. In all these cases, the billiard region is the fundamental domain of a Lorentzian Coxeter group (“Coxeter billiard”). Furthermore, the Coxeter group in question is crystallographic and turns out to be the Weyl group of a Lorentzian Kac-Moody algebra. The billiard table is then the fundamental Weyl chamber of a Lorentzian Kac-Moody algebra [45, 46] and the billiard is also called a “Kac-Moody billiard”. This enables one to reformulate the dynamics as a motion in the Cartan subalgebra of the Lorentzian Kac-Moody algebra, hinting at the potential — and still conjectural at this stage — existence of a deeper, infinite-dimensional symmetry of the theory.

The purpose of this section is threefold:
  1. 1.

    First, we exhibit other theories besides pure gravity in four dimensions which also lead to a Coxeter billiard. We stress further how exceptional these theories are in the space of all theories described by the action Equation (2.1).

     
  2. 2.

    Second, we show how to reformulate the dynamics as a motion in the Cartan subalgebra of a Lorentzian Kac-Moody algebra.

     
  3. 3.

    Finally, we connect the Lorentzian Kac-Moody algebra that appears in the BKL-limit to the “hidden” symmetry group \({{\mathcal U}_3}\) in the simplest case when the real Lie algebra \({{\mathfrak u}_3}\) of the group \({{\mathcal U}_3}\) is the split real form of the corresponding complexified Lie algebra \({\mathfrak u}_3^{\mathbb C}\). (These concepts will be defined below.) The general case will be dealt with in Section 7, after we have recalled the most salient features of the theory of real forms in Section 6.

     

5.1 More on Coxeter billiards

5.1.1 The Coxeter billiard of pure gravity in D spacetime dimensions

We start by providing other examples of theories leading to regular billiards, focusing first on pure gravity in any number of D (> 3) spacetime dimensions. In this case, there are d = D − 1 scale factors β i and the relevant walls are the symmetry walls, Equation (2.48),
$${s_i}(\beta) \equiv {\beta ^{i + 1}} - {\beta ^i} = 0\qquad (i = 1,2, \cdots ,d - 1),$$
(5.1)
and the curvature wall, Equation (2.49),
$$r(\beta) \equiv 2{\beta ^1} + {\beta ^2} + \cdots + {\beta ^{d - 2}} = 0.$$
(5.2)
There are thus d relevant walls, which define a simplex in (d − 1)-dimensional hyperbolic space The scalar products of the linear forms defining these walls are easily computed. One finds as non-vanishing products
$$\begin{array}{*{20}c}{({s_i}\vert {s_i}) = 2\qquad (i = 1, \cdots ,d - 1),} \\{(r\vert r) = 2,\quad \quad \quad \quad \quad \quad \quad \;} \\{({s_{i + 1}}\vert {s_i}) = - 1\qquad (i = 2, \cdots ,d - 1)} \\{(r\vert {s_1}) = - 1,\quad \quad \quad \quad \quad \quad \quad} \\{(r\vert {s_{d - 2}}) = - 1.\quad \quad \quad \quad \quad \quad \quad \;\;\;} \\\end{array}$$
(5.3)
The matrix of the scalar products of the wall forms is thus the Cartan matrix of the (simply-laced) Lorentzian Kac-Moody algebra \(A_{d - 2}^{+ +}\) with Dynkin diagram as in Figure 24. The roots of the underlying finite-dimensional algebra Ad−2 are given by s i (i = 1, ⋯, d − 3) and r. The affine root is sd−2 and the overextended root is sd−1.
Figure 24

The Dynkin diagram of the hyperbolic Kac-Moody algebra \(A_{d - 2}^{+ +}\) which controls the billiard dynamics of pure gravity in D = d + 1 dimensions. The nodes s1, ⋯, sd−1 represent the “symmetry walls” arising from the off-diagonal components of the spatial metric, and the node r corresponds to a “curvature wall” coming from the spatial curvature. The horizontal line is the Dynkin diagram of the underlying Ad−2-subalgebra and the two topmost nodes, sd−2 and sd−1, give the affine- and overextension, respectively.

Accordingly, in the case of pure gravity in any number of spacetime dimensions, one finds also that the billiard region is regular. This provides new examples of Coxeter billiards, with Coxeter groups \(A_{d - 2}^{+ +}\), which are also Kac-Moody billiards since the Coxeter groups are the Weyl groups of the Kac-Moody algebras \(A_{d - 2}^{+ +}\).

5.1.2 The Coxeter billiard for the coupled gravity-3-Form system Coxeter polyhedra

Let us review the conditions that must be fulfilled in order to get a Kac-Moody billiard and let us emphasize how restrictive these conditions are. The billiard region computed from any theory coupled to gravity with n dilatons in D = d + 1 dimensions always defines a convex polyhedron in a (d + n − 1)-dimensional hyperbolic space \({{\mathcal H}_{d-1}}\). In the general case, the dihedral angles between adjacent faces of \({{\mathcal H}_{d+n-1}}\) can take arbitrary continuous values, which depend on the dilaton couplings, the spacetime dimensions and the ranks of the p-forms involved. However, only if the dihedral angles are integer submultiples of π (meaning of the form π/k for k ∈ ℤ≥2) do the reflections in the faces of \({{\mathcal H}_{d+n-1}}\) define a Coxeter group. In this special case the polyhedron is called a Coxeter polyhedron. This Coxeter group is then a (discrete) subgroup of the isometry group of \({{\mathcal H}_{d+n-1}}\).

In order for the billiard region to be identifiable with the fundamental Weyl chamber of a Kac-Moody algebra, the Coxeter polyhedron should be a simplex, i.e., bounded by d + n walls in a d + n − 1-dimensional space. In general, the Coxeter polyhedron need not be a simplex.

There is one additional condition. The angle ϑ between two adjacent faces i and j is given, in terms of the Coxeter exponents, by
$$\vartheta = {\pi \over {{m_{ij}}}}{.}$$
(5.4)
Coxeter groups that correspond to Weyl groups of Kac-Moody algebras are the crystallographic Coxeter groups for which m ij ∈ {2, 3, 4, 6, ∞}. So, the requirement for a gravitational theory to have a Kac-Moody algebraic description is not just that the billiard region is a Coxeter simplex but also that the angles between adjacent walls are such that the group of reflections in these walls is crystallographic.

These conditions are very restrictive and hence gravitational theories which can be mapped to a Kac-Moody algebra in the BKL-limit are rare.

5.1.2.1 The Coxeter billiard of eleven-dimensional supergravity

Consider for instance the action (2.1) for gravity coupled to a single three-form in D = d + 1 spacetime dimensions. We assume D ≥ 6 since in lower dimensions the 3-form is equivalent to a scalar (D = 5) or has no degree of freedom (D < 5).

Theorem: Whenever a p-form (p ≥ 1) is present, the curvature wall is subdominant as it can be expressed as a linear combination with positive coefficients of the electric and magnetic walls of the p-forms. (These walls are all listed in Section 2.5.)

Proof: The dominant electric wall is (assuming the presence of a dilaton)
$${e_{1 \cdots p}}(\beta) \equiv {\beta ^1} + {\beta ^2} + \cdots + {\beta ^p} - {{{\lambda _p}} \over 2}\phi = 0,$$
(5.5)
while one of the magnetic wall reads
$${m_{1,p + 1, \cdots ,d - 2}}(\beta) \equiv {\beta ^1} + {\beta ^{p + 1}} + \cdots + {\beta ^{d - 2}} + {{{\lambda _p}} \over 2}\phi = 0,$$
(5.6)
so that the dominant curvature wall is just the sum e1⋯p(β) + m1,p+1, ⋯, d−2(β).
It follows that in the case of gravity coupled to a single three-form in D = d + 1 spacetime dimensions, the relevant walls are the symmetry walls, Equation (2.48),
$${s_i}(\beta) \equiv {\beta ^{i + 1}} - {\beta ^i} = 0,\qquad i = 1,2, \cdots ,d - 1$$
(5.7)
(as always) and the electric wall
$${e_{123}}(\beta) \equiv {\beta ^1} + {\beta ^2} + {\beta ^3} = 0$$
(5.8)
(D ≥ 8) or the magnetic wall
$${m_{1 \cdots D - 5}}(\beta) \equiv {\beta ^1} + {\beta ^2} + \cdots {\beta ^{D - 5}} = 0$$
(5.9)
(D ≤ 8). Indeed, one can express the magnetic walls as linear combinations with (in general non-integer) positive coefficients of the electric walls for D ≥ 8 and vice versa for D ≤ 8. Hence the billiard table is always a simplex (this would not be true had one a dilaton and various forms with different dilaton couplings).
However, it is only for D = 11 that the billiard is a Coxeter billiard. In all the other spacetime dimensions, the angle between the relevant p-form wall and the symmetry wall that does not intersect it orthogonally is not an integer submultiple of π. More precisely, the angle between
  • the magnetic wall β1 and the symmetry wall β2β1 (D = 6),

  • the magnetic wall β1 + β2 and the symmetry wall β3β2 (D = 7), and

  • the electric wall β1 + β2 + β3 and the symmetry wall β4β3 (D ≥ 8),

is easily verified to be an integer submultiple of π only for D = 11, for which it is equal to π/3. From the point of view of the regularity of the billiard, the spacetime dimension D = 11 is thus privileged. This is of course also the dimension privileged by supersymmetry. It is quite intriguing that considerations a priori quite different (billiard regularity on the one hand, supersymmetry on the other hand) lead to the same conclusion that the gravity-3-form system is quite special in D = 11 spacetime dimensions.
For completeness, we here present the wall system relevant for the special case of D = 11. We obtain ten dominant wall forms, which we rename α1, ⋯, α10,
$$\begin{array}{*{20}c}{{\alpha _m}(\beta) = {\beta ^{m + 1}} - {\beta ^m}\qquad (m = 1, \cdots ,10),} \\{{\alpha _{10}}(\beta)={\beta ^1}{\rm{+}}{\beta ^2}{\rm{+}}{\beta ^3}.\quad \quad \quad \quad \quad \quad \quad} \\\end{array}$$
(5.10)
then, defining a new collective index i = (m, 10), we see that the scalar products between these wall forms can be organized into the matrix
$${A_{ij}} = 2{{({\alpha _i}\vert {\alpha _j})} \over {({\alpha _i}\vert {\alpha _i})}} = \left({\begin{array}{*{20}c}2 & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{- 1} & 2 & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & {- 1} & 2 & {- 1} & 0 & 0 & 0 & 0 & 0 & {- 1} \\0 & 0 & {- 1} & 2 & {- 1} & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & {- 1} & 2 & {- 1} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & {- 1} & 2 & {- 1} & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & {- 1} & 2 & {- 1} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & {- 1} & 2 & {- 1} & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & {- 1} & 2 & 0 \\0 & 0 & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\\end{array}} \right),$$
(5.11)
which can be identified with the Cartan matrix of the hyperbolic Kac-Moody algebra E10 that we have encountered in Section 4.10.2. We again display the corresponding Dynkin diagram in Figure 25, where we point out the explicit relation between the simple roots and the walls of the Einstein-3-form theory. It is clear that the nine dominant symmetry wall forms correspond to the simple roots α m of the subalgebra \({\mathfrak {sl}(10,\,\mathbb R)}\). The enlargement to E10 is due to the tenth exceptional root realized here through the dominant electric wall form e123.
Figure 25

The Dynkin diagram of E10. Labels m = 1, ⋯, 9 enumerate the nodes corresponding to simple roots, α m , of the \({\mathfrak {sl}(10,\,\mathbb R)}\) subalgebra and the exceptional node, labeled “10”, is associated to the electric wall α10 = e123.

5.2 Dynamics in the Cartan subalgebra

We have just learned that, in some cases, the group of reflections that describe the (possibly chaotic) dynamics in the BKL-limit is a Lorentzian Coxeter group \({\mathfrak C}\). In this section we fully exploit this algebraic fact and show that whenever \({\mathfrak C}\) is crystallographic, the dynamics takes place in the Cartan subalgebra \({\mathfrak h}\) of the Lorentzian Kac-Moody algebra \({\mathfrak g}\), for which \({\mathfrak C}\) is the Weyl group. Moreover, we show that the “billiard table” can be identified with the fundamental Weyl chamber in \({\mathfrak h}\).

5.2.1 Billiard dynamics in the Cartan subalgebra

5.2.1.1 Scale factor space and the wall system
Let us first briefly review some of the salient features encountered so far in the analysis. In the following we denote by \({{\mathcal M}_\beta}\) the Lorentzian “scale factor”-space (or β-space) in which the billiard dynamics takes place. Recall that the metric in \({{\mathcal M}_\beta}\), induced by the Einstein-Hilbert action, is a flat Lorentzian metric, whose explicit form in terms of the (logarithmic) scale factors reads
$${G_{\mu \nu}}\,d{\beta ^\mu}\,d{\beta ^\nu} = \sum\limits_{i = 1}^d d {\beta ^i}\,d{\beta ^i} - \left({\sum\limits_{i = 1}^d d {\beta ^i}} \right)\left({\sum\limits_{j = 1}^d d {\beta ^j}} \right) + d\phi \,d\phi ,$$
(5.12)
where d counts the number of physical spatial dimensions (see Section 2.5). The role of all other “off-diagonal” variables in the theory is to interrupt the free-flight motion of the particle, by adding walls in \({{\mathcal M}_\beta}\) that confine the motion to a limited region of scale factor space, namely a convex cone bounded by timelike hyperplanes. When projected onto the unit hyperboloid, this region defines a simplex in hyperbolic space which we refer to as the “billiard table”.
One has, in fact, more than just the walls. The theory provides these walls with a specific normalization through the Lagrangian, which is crucial for the connection to Kac-Moody algebras. Let us therefore discuss in somewhat more detail the geometric properties of the wall system. The metric, Equation (5.12), in scale factor space can be seen as an extension of a flat Euclidean metric in Cartesian coordinates, and reflects the Lorentzian nature of the vector space \({{\mathcal M}_\beta}\). In this space we may identify a pair of coordinates (β i , φ) with the components of a vector \(\beta \in {{\mathcal M}_\beta}\), with respect to a basis {ū μ } of \({{\mathcal M}_\beta}\), such that
$${\bar u_\mu}\cdot{\bar u_\nu} = {G_{\mu \nu}}.$$
(5.13)
The walls themselves are then defined by hyperplanes in this linear space, i.e., as linear forms ω = ω μ σ μ , for which ω = 0, where {σ μ } is the basis dual to {ū μ }. The pairing ω(β) between a vector \(\beta \in {{\mathcal M}_\beta}\) and a form \(\omega \in {\mathcal M}_\beta ^{\star}\) is sometimes also denoted by 〈ω, β〉, and for the two dual bases we have, of course,
$$\langle {{{\underline \sigma}^\mu},{{\bar u}_\nu}} \rangle = \delta _\nu ^\mu .$$
(5.14)
We therefore find that the walls can be written as linear forms in the scale factors:
$$\omega (\beta) = \sum\limits_{\mu ,\nu} {{\omega _\mu}} {\beta ^\nu}\langle {{{\underline \sigma}^\mu},{{\bar u}_\nu}} \rangle = \sum\limits_\mu {{\omega _\mu}} {\beta ^\mu} = \sum\limits_{i = 1}^d {{\omega _i}} {\beta ^i} + {\omega _\phi}\phi .$$
(5.15)
We call ω(β) wall forms. With this interpretation they belong to the dual space \({\mathcal M}_\beta ^{\star}\), i.e.,
$$\begin{array}{*{20}c}{\mathcal{M}_\beta ^ \star \ni \omega :{\mathcal{M}_\beta} \rightarrow \mathbb{R}\quad \quad \quad \quad \quad \quad} \\{\beta \mapsto \omega (\beta).} \\ \end{array}$$
(5.16)
From Equation (5.16) we may conclude that the walls bounding the billiard are the hyperplanes ω = 0 through the origin in \({{\mathcal M}_\beta}\) which are orthogonal to the vector with components ω μ = G μv ω v .
It is important to note that it is the wall forms that the theory provides, as arguments of the exponentials in the potential, and not just the hyperplanes on which these forms ω vanish. The scalar products between the wall forms are computed using the metric in the dual space \({\mathcal M}_\beta ^{\star}\), whose explicit form was given in Section 2.5,
$$(\omega \vert \omega {\prime}) \equiv {G^{\mu \nu}}{\omega _\mu}{\omega _\nu} = \sum\limits_{i = 1}^d {{\omega _i}} {\omega _i}\prime - {1 \over {d - 1}}\left({\sum\limits_{i = 1}^d {{\omega _i}}} \right)\left({\sum\limits_{j = 1}^d {{\omega _j}\prime}} \right) + {\omega _\phi}{\omega _\phi}\prime ,\qquad \omega ,\omega {\prime} \in {\mathcal{M}_\beta}.$$
(5.17)
5.2.1.2 Scale factor space and the Cartan subalgebra
The crucial additional observation is that (for the “interesting” theories) the matrix A associated with the relevant walls ω A ,
$${A_{AB}} = 2{{({\omega _A}\vert {\omega _B})} \over {({\omega _A}\vert {\omega _A})}}$$
(5.18)
is a Cartan matrix, i.e., besides having 2’s on its diagonal, which is rather obvious, it has as off-diagonal entries non-positive integers (with the property A AB ≠ 0 ⇒ A BA ≠ 0). This Cartan matrix is of course symmetrizable since it derives from a scalar product.

For this reason, one can usefully identify the space of the scale factors with the Cartan subalgebra \(\mathfrak{h}\) of the Kac-Moody algebra \(\mathfrak{g}{(A)}\) defined by A. In that identification, the wall forms become the simple roots, which span the vector space \(\mathfrak{h}^{\star}={\rm span}\{{\alpha _1},\, \cdots \,,\,{\alpha _r}\}\) dual to the Cartan subalgebra. The rank r of the algebra is equal to the number of scale factors β μ , including the dilaton(s) if any ((β μ ) ≡ (β i , φ)). This number is also equal to the number of walls since we assume the billiard to be a simplex. So, both A and μ run from 1 to r. The metric in \({{\mathcal M}_\beta}\), Equation (5.12), can be identified with the invariant bilinear form of \(\mathfrak{g}\), restricted to the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{g}\). The scale factors β μ of \(\mathcal{M}_{\beta}\) become then coordinates h μ on the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{g}(A)\).

The Weyl group of a Kac-Moody algebra has been defined first in the space \(\mathfrak{h}^{\star}\) as the group of reflections in the walls orthogonal to the simple roots. Since the metric is non degenerate, one can equivalently define by duality the Weyl group in the Cartan algebra \(\mathfrak{h}\) itself (see Section 4.7). For each reflection r i on \(\mathfrak{h}^{\star}\) we associate a dual reflection \(r_i^ \vee\) as follows,
$$r_i^ \vee (\beta) = \beta - \langle {{\alpha _i},\beta} \rangle \alpha _i^ \vee ,\qquad \beta ,\alpha _i^ \vee \in \mathfrak{h},$$
(5.19)
which is the reflection relative to the hyperplane α i (β) = 〈α i , β〉 = 0. This expression can be rewritten (see Equation (4.59)),
$$r_i^ \vee (\beta) = \beta - {{2(\beta \vert \alpha _i^ \vee)} \over {(\alpha _i^ \vee \vert \alpha _i^ \vee)}}\alpha _i^ \vee ,$$
(5.20)
or, in terms of the scale factor coordinates β μ ,
$${\beta ^\mu} \rightarrow {\beta^\mu}\prime = {\beta ^\mu} - {{2(\beta \vert {\omega ^ \vee})} \over {({\omega ^ \vee}\vert {\omega ^ \vee})}}{\omega ^{\vee \mu}}.$$
(5.21)
This is precisely the billiard reflection Equation (2.45) found in Section 2.4.
Thus, we have the following correspondence:
$$\begin{array}{*{20}c}{{\mathcal{M}_\beta} \equiv \mathfrak{h},\quad \quad \quad \;} \\{\mathcal{M}_\beta ^ \star \equiv {\mathfrak{h}^ \star},\quad \quad \quad \;} \\{{\omega _A}(\beta) \equiv {\alpha _A}(h),\quad \quad \;\;\;} \\{{\rm{billiard\; wall\; reflections}} \equiv {\rm{fundamental\; Weyl\; reflections}}.\;\;} \\ \end{array}$$
(5.22)
As we have also seen, the Kac-Moody algebra \(\mathfrak{g}(A)\) is Lorentzian since the signature of the metric Equation (5.12) is Lorentzian. This fact will be crucial in the analysis of subsequent sections and is due to the presence of gravity, where conformal rescalings of the metric define timelike directions in scale factor space.
We thereby arrive at the following important result [45, 46, 48]:

The dynamics of (a restricted set of) theories coupled to gravity can in the BKL-limit be mapped to a billiard motion in the Cartan subalgebra \(\mathfrak{h}\) of a Lorentzian Kac-Moody algebra \(\mathfrak{g}\).

5.2.2 The fundamental Weyl chamber and the billiard table

Let \(\mathcal{B}_{\mathcal{M}_{\beta}}\) denote the region in scale factor space to which the billiard motion is confined,
$${\mathcal{B}_{{\mathcal{M}_\beta}}} = \{\beta \in {\mathcal{M}_\beta}\,\vert \,{\omega _A}(\beta) \geq 0\} ,$$
(5.23)
where the index A runs over all relevant walls. On the algebraic side, the fundamental Weyl chamber in \(\mathfrak{h}\) is the closed convex (half) cone given by
$${\mathcal{W}_\mathfrak{h}} = \{h \in \mathfrak{h}\vert{\alpha _A}(h) \geq 0;\,A = 1, \cdots ,{\rm{rank}}\;\mathfrak{g}\} .$$
(5.24)
We see that the conditions α A (h) ≥ 0 defining \(\mathcal{W}_{\mathfrak{h}}\) are equivalent, upon examination of Equation (5.22), to the conditions ω A (β) ≥ 0 defining the billiard table \({{\mathcal B}_{{{\mathcal M}_\beta}}}\).
We may therefore make the crucial identification
$${\mathcal{W}_\mathfrak{h}} \equiv{\mathcal{B}_{{\mathcal{M}_\beta}}},$$
(5.25)
which means that the particle geodesic is confined to move within the fundamental Weyl chamber of \(\mathfrak{h}\). From the billiard analysis in Section 2 we know that the piecewise motion in scale-factor space is controlled by geometric reflections with respect to the walls ω A (β) = 0. By comparing with the dominant wall forms and using the correspondence in Equation (5.22) we may further conclude that the geometric reflections of the coordinates β μ (τ) are controlled by the Weyl group in the Cartan subalgebra of \(\mathfrak{g}(A)\).

5.2.3 Hyperbolicity implies chaos

We have learned that the BKL dynamics is chaotic if and only if the billiard table is of finite volume when projected onto the unit hyperboloid. From our discussion of hyperbolic Coxeter groups in Section 3.5, we see that this feature is equivalent to hyperbolicity of the corresponding Kac-Moody algebra. This leads to the crucial statement [45, 46, 48]:

If the billiard region of a gravitational system in the BKL-limit can be identified with the fundamental Weyl chamber of a hyperbolic Kac-Moody algebra, then the dynamics is chaotic.

As we have also discussed above, hyperbolicity can be rephrased in terms of the fundamental weights Λ i defined as
$$\langle {{\Lambda _j},\alpha _i^ \vee} \rangle = {{2({\Lambda _j}\vert {\alpha _i})} \over {({\alpha _i}\vert {\alpha _i})}} \equiv {\delta _{ij}},\qquad \alpha _i^ \vee \in \mathfrak{h},{\Lambda _i} \in {\mathfrak{h}^ \star}.$$
(5.26)
Just as the fundamental Weyl chamber in \(\mathfrak{h}^{\star}\) can be expressed in terms of the fundamental weights (see Equation (3.40)), the fundamental Weyl chamber in \(\mathfrak{h}\) can be expressed in a similar fashion in terms of the fundamental coweights:
$${\mathcal{W}_\mathfrak{h}} = \{\beta \in \mathfrak{h}\vert \beta = \sum\limits_i {{a_i}} \Lambda _i^ \vee ,\,{a_i} \in {\mathbb{R}_{\geq 0}}\} {.}$$
(5.27)
As we have seen (Sections 3.5 and 4.8), hyperbolicity holds if and only if none of the fundamental weights are spacelike,
$$({\Lambda _i}\vert {\Lambda _i}) \leq 0,$$
(5.28)
for all \(i \in \{1,\, \cdots \,,\,{\rm{rank}}\,\mathfrak{g}\}\).
5.2.3.1 Example: Pure gravity in D = 3 + 1 and \(A_1^{+ +}\)

Let us return once more to the example of pure four-dimensional gravity, i.e., the original “BKL billiard”. We have already found in Section 3 that the three dominant wall forms give rise to the Cartan matrix of the hyperbolic Kac-Moody algebra \(A_1^{+ +}\) [46, 48]. Since the algebra is hyperbolic, this theory exhibits chaotic behavior. In this example, we verify that the Weyl chamber is indeed contained within the lightcone by computing explicitly the norms of the fundamental weights.

It is convenient to first write the simple roots in the β-basis as follows¿
$$\begin{array}{*{20}c}{{\alpha _1} = (2,0,0)\;\;} \\{{\alpha _2} = (- 1,1,0)\;} \\{{\alpha _3} = (0, - 1,1).} \\ \end{array}$$
(5.29)
Since the Cartan matrix of \(A_1^{+ +}\) is symmetric, the relations defining the fundamental weights are
$$({\alpha _i}\vert {\Lambda _j}) \equiv {\delta _{ij}}.$$
(5.30)
By solving these equations for Λ i we deduce that the fundamental weights are
$$\begin{array}{*{20}c}{{\Lambda _1} = - {3 \over 2}{\alpha _1} - 2{\alpha _2} - {\alpha _3} = (- 1, - 1, - 1),} \\{{\Lambda _2} = - 2{\alpha _1} - 2{\alpha _2} - 2{\alpha _3} = (0,1, - 1),\quad} \\ {{\Lambda _3} = - {\alpha _1} - {\alpha _2} = (- 1, - 1,0),\quad \quad \quad \;\;\;} \\ \end{array}$$
(5.31)
where in the last step we have written the fundamental weights in the β-basis. The norms may now be computed with the metric in root space and become
$$({\Lambda _1}\vert {\Lambda _1}) = - {3 \over 2},\qquad ({\Lambda _2}\vert {\Lambda _2}) = - 2,\qquad ({\Lambda _3}\vert {\Lambda _3}) = 0{.}$$
(5.32)
We see that Λ1 and Λ2 are timelike and that Λ3 is lightlike. Thus, the Weyl chamber is indeed contained inside the lightcone, the algebra is hyperbolic and the billiard is of finite volume, in agreement with what we already found [46].

5.3 Understanding the emerging Kac-Moody algebra

We shall now relate the Kac-Moody algebra whose fundamental Weyl chamber emerges in the BKL-limit to the U-duality group that appears upon toroidal dimensional reduction to three spacetime dimensions. We shall do this first in the case when \(\mathfrak{u}_3\) is a split real form. By this we mean that the real algebra \(\mathfrak{u}_3\) possesses the same Chevalley-Serre presentation as \(\mathfrak{u}_3^{\mathbb{C}}\), but with coefficients restricted to be real numbers. This restriction is mathematically consistent because the coefficients appearing in the Chevalley-Serre presentation are all reals (in fact, integers).

The fact that the billiard structure is preserved under reduction turns out to be very useful for understanding the emergence of “overextended” algebras in the BKL-limit. By computing the billiard in three spacetime dimensions instead of in maximal dimension, the relation to U-duality groups becomes particularly transparent and the computation of the billiard follows a similar pattern for all cases. We will see that if \(\mathfrak{u}_3\) is the algebra representing the internal symmetry of the non-gravitational degrees of freedom in three dimensions, then the billiard is controlled by the Weyl group of the overextended algebra \(\mathfrak{u}_3^{++}\). The analysis is somewhat more involved when \(\mathfrak{u}_3\) is non-split, and we postpone a discussion of this until Section 7.

5.3.1 Invariance under toroidal dimensional reduction

It was shown in [41] that the structure of the billiard for any given theory is completely unaffected by dimensional reduction on a torus. In this section we illustrate this by an explicit example rather than in full generality. We consider the case of reduction of eleven-dimensional supergravity on a circle.

The compactification ansatz in the conventions of [35, 41] is
$${{\rm{g}}_{MN}} = \left({\begin{array}{*{20}c}{{e^{- 2({4 \over {3\sqrt 2}}\hat{\varphi})}}\quad \quad \quad \quad {e^{- 2({4 \over {3\sqrt 2}}\hat{\varphi})}}{{\hat{\mathcal{A}}}_\nu}\quad \quad \quad \;\;} \\ {{e^{- 2({4 \over {3\sqrt 2}}\hat{\varphi})}}{{\hat{\mathcal{A}}}_\mu}\quad {e^{- 2({-1 \over {6\sqrt 2}}\hat{\varphi})}}{{{\rm{\hat g}}}_{\mu \nu}} + {e^{- 2({4 \over {3\sqrt 2}}\hat{\varphi})}}{{\hat{\mathcal{A}}}_\mu}{{\hat{\mathcal{A}}}_\nu}} \\ \end{array}} \right),$$
(5.33)
where μ, v = 0, 2, ⋯, 10, i.e., the compactification is performed along the first spatial direction16. We will refer to the new lower-dimensional fields \(\hat \varphi\) and \({\hat {\mathcal A}_\mu}\) as the dilaton and the Kaluza-Klein (KK) vector, respectively. Quite generally, hatted fields are low-dimensional fields. The ten-dimensional Lagrangian becomes
$$\begin{array}{*{20}c} {\mathcal{L}_{(10)}^{{\rm{SUGR}}{{\rm{A}}_{11}}} = {R_{(10)}} \star {\bf{1}} - \star d\hat{\varphi}\wedge d\hat{\varphi} - {1 \over 2}{e^{- 2({3 \over {2\sqrt 2}}\hat{\varphi})}} \star {{\hat{\mathcal{F}}}^{(2)}} \wedge {{\hat{\mathcal{F}}}^{(2)}}\quad \quad \quad \quad \quad} \\{- {1 \over 2}{e^{- 2({1 \over {2\sqrt 2}}\hat{\varphi})}} \star {{\hat{F}}^{(4)}} \wedge {{\hat{F}}^{(4)}} - {1 \over 2}{e^{- 2({{- 1} \over {\sqrt 2}}\hat{\varphi})}} \star {{\hat{F}}^{(3)}} \wedge {{\hat{F}}^{(3)}},} \\ \end{array}$$
(5.34)
where \(\hat{\mathcal{F}}^{(2)}=d\hat{\mathcal{A}}^{(1)}\) and \({\hat F^{(4)}},\,{\hat F^{(3)}}\) are the field strengths in ten dimensions originating from the eleven-dimensional 3-form field strength F(4) = dA(3).
Examining the new form of the metric reveals that the role of the scale factor β1, associated to the compactified dimension, is now instead played by the ten-dimensional dilaton, \(\hat \varphi\). Explicitly we have
$${\beta ^1} = {4 \over {3\sqrt 2}}\hat{\varphi} .$$
(5.35)
The nine remaining eleven-dimensional scale factors, β2, ⋯, β10, may in turn be written in terms of the new scale factors, \({\hat \beta ^a}\), associated to the ten-dimensional metric, ĝ μv , and the dilaton in the following way:
$${\beta ^a} = {\hat \beta ^a} - {1 \over {6\sqrt 2}}\hat{\varphi} \qquad (a = 2, \cdots ,10).$$
(5.36)
We are interested in finding the dominant wall forms in terms of the new scale factors \({\hat \beta _2},\, \cdots \,,\,{\hat \beta _{10}}\) and \(\hat \varphi\). It is clear that we will have eight ten-dimensional symmetry walls,
$${\hat s_{\hat m}}(\hat \beta) = {\hat \beta ^{\hat m + 1}} - {\hat \beta ^{\hat m}}\qquad (\hat m = 2, \cdots ,9),$$
(5.37)
which correspond to the eight simple roots of \(\mathfrak{sl}(9,\,\mathbb{R})\). Using Equation (5.35) and Equation (5.36) one may also check that the symmetry wall β2β 1 , that was associated with the compactified direction, gives rise to an electric wall of the Kaluza-Klein vector,
$$\hat e_2^{\hat{\mathcal{A}}}(\hat \beta) = {\hat \beta ^2} - {3 \over {2\sqrt 2}}\hat{\varphi} .$$
(5.38)
The metric in the dual space gets modified in a natural way,
$$({\hat \alpha _k}\vert {\hat \alpha _l}) = \sum\limits_{i = 2}^{10} {{{\hat \alpha}_{ki}}} {\hat \alpha _{li}} - {1 \over 8}\left({\sum\limits_{i = 2}^{10} {{{\hat \alpha}_{ki}}}} \right)\left({\sum\limits_{j = 2}^{10} {{{\hat \alpha}_{lj}}}} \right) + {\hat \alpha _{k\hat{\varphi}}}{\hat \alpha _{l\hat{\varphi}}},$$
(5.39)
i.e., the dilaton contributes with a flat spatial direction. Using this metric it is clear that \(\hat e_2^{\hat {\mathcal A}}\) has non-vanishing scalar product only with the second symmetry wall \({\hat s_2} = {\hat \beta ^3} - {\hat \beta ^2},\,(\hat e_2^{\hat {\mathcal A}}\vert{\hat s_2}) = - 1\), and it follows that the electric wall of the Kaluza-Klein vector plays the role of the first simple root of \(\mathfrak{sl}(10,\,\mathbb{R}),\,{\hat \alpha _1} \equiv \hat e_2^{\hat {\mathcal A}}\). The final wall form that completes the set will correspond to the exceptional node labeled “10” in Figure 25 and is now given by one of the electric walls of the NS-NS 2-form Â(2), namely
$${\hat \alpha _{10}} \equiv \hat e_{23}^{{{\hat A}^{(2)}}}(\hat \beta) = {\hat \beta ^2} + {\hat \beta ^3} + {1 \over {\sqrt 2}}\hat{\varphi}.$$
(5.40)
It is then easy to verify that this wall form has non-vanishing scalar product only with the third simple root \({\hat \alpha _3} = {\hat s_3},\,(\hat e_{23}^{{{\hat A}^{(2)}}}\vert{\hat s_3}) = - 1\), as desired.
We have thus shown that the E10 structure is sufficiently rigid to withstand compactification on a circle with the new simple roots explicitly given by
$$\{{\hat{\alpha _1}},{\hat{\alpha _2}}, \cdots ,{\hat{\alpha _9}},{\hat {\alpha _{10}}}\} = \{\hat{e}_2^{\hat{\mathcal{A}}},{\hat s_2}, \cdots ,{\hat{s}_9},\hat{e}_{23}^{{{\hat A}^{(2)}}}\} {.}$$
(5.41)
This result is in fact true also for the general case of compactification on tori, T n . When reaching the limiting case of three dimensions, all the non-gravity wall forms correspond to the electric and magnetic walls of the axionic scalars. We will discuss this case in detail below.

For non-toroidal reductions the above analysis is drastically modified [166, 165]. The topology of the internal manifold affects the dominant wall system, and hence the algebraic structure in the lower-dimensional theory is modified. In many cases, the billiard of the effective compactified theory is described by a (non-hyperbolic) regular Lorentzian subalgebra of the original hyperbolic Kac-Moody algebra [98].

The walls are also invariant under dualization of a p-form into a (Dp − 2)-form; this simply exchanges magnetic and electric walls.

5.3.2 Iwasawa decomposition for split real forms

We will now exploit the invariance of the billiard under dimensional reduction, by considering theories that — when compactified on a torus to three dimensions — exhibit “hidden” internal global symmetries \({{\mathcal U}_3}\). By this we mean that the three-dimensional reduced theory is described, after dualization of all vectors to scalars, by the sum of the Einstein-Hilbert Lagrangian coupled to the Lagrangian for the nonlinear sigma model \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\). Here, \({\mathcal K}({{\mathcal U}_3})\) is the maximal compact subgroup defining the “local symmetries”. In order to understand the connection between the U-duality group \({{\mathcal U}_3}\) and the Kac-Moody algebras appearing in the BKL-limit, we must first discuss some important features of the Lie algebra \(\mathfrak{u}_3\).

Let \(\mathfrak{u}_3\) be a split real form, meaning that it can be defined in terms of the Chevalley-Serre presentation of the complexified Lie algebra \(\mathfrak{u}_3^{\mathbb{C}}\) by simply restricting all linear combinations of generators to the real numbers ℝ. Let \(\mathfrak{h}_3\) be the Cartan subalgebra of \(\mathfrak{u}_3\) appearing in the Chevalley-Serre presentation, spanned by the generators \(\alpha _1^ \vee, \, \cdots \,,\,\alpha _n^ \vee\). It is maximally noncompact (see Section 6). An Iwasawa decomposition of \(\mathfrak{u}_3\) is a direct sum of vector spaces of the following form,
$$\mathfrak{u}_3=\mathfrak{k}_3\oplus \mathfrak{h}_3\oplus \mathfrak{n}_3,$$
(5.42)
where \(\mathfrak{k}_3\) is the “maximal compact subalgebra” of \(\mathfrak{u}_3\), and \(\mathfrak{n}_3\) is the nilpotent subalgebra spanned by the positive root generators E α , ∀α ∈ Δ+.
The corresponding Iwasawa decomposition at the group level enables one to write uniquely any group element as a product of an element of the maximally compact subgroup times an element in the subgroup whose Lie algebra is \(\mathfrak{h}_3\) times an element in the subgroup whose Lie algebra is \(\mathfrak{n}_3\). An arbitrary element \({\mathcal V}(x)\) of the coset \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\) is defined as the set of equivalence classes of elements of the group modulo elements in the maximally compact subgroup. Using the Iwasawa decomposition, one can go to the “Borel gauge”, where the elements in the coset are obtained by exponentiating only generators belonging to the Borel subalgebra,
$$\mathfrak{b}_3=\mathfrak{h}_3\oplus \mathfrak{n}_3\subset \mathfrak{u}_3.$$
(5.43)
In that gauge we have
$$\mathcal{V}(x) = {\rm{Exp}}\left[ {\phi (x)\cdot{\mathfrak{h}_3}} \right]\,{\rm{Exp}}\left[ {\chi (x)\cdot{\mathfrak{n}_3}} \right],$$
(5.44)
where φ and χ are (sets of) coordinates on the coset space \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\). A Lagrangian based on this coset will then take the generic form (see Section 9)
$${\mathcal{L}_{{\mathcal{U}_3}/\mathcal{K}({\mathcal{U}_3})}} = \sum\limits_{i = 1}^{\dim {\mathfrak{h}_3}} {\partial _x}{\phi ^{(i)}}(x){\partial _x}{\phi ^{(i)}}(x) + \sum\limits_{\alpha \in {\Delta _ +}} {{e^{2\alpha (\phi)}}} \left[ {{\partial _x}{\chi ^{(\alpha)}}(x) + \cdots} \right]\left[ {{\partial _x}{\chi ^{(\alpha)}}(x) + \cdots} \right],$$
(5.45)
where x denotes coordinates in spacetime and the “ellipses” denote correction terms that are of no relevance for our present purposes. We refer to the fields {φ} collectively as dilatons and the fields {χ} as axions. There is one axion field χ(α) for each positive root α ∈ Δ+ and one dilaton field φ(i) for each Cartan generator \(\alpha _i^ \vee \in \mathfrak{h}_3\).

The Lagrangian (5.45) coupled to the pure three-dimensional Einstein-Hilbert term is the key to understanding the appearance of the Lorentzian Coxeter group \(\mathfrak{u}_3^{++}\) in the BKL-limit.

5.3.3 Starting at the bottom — Overextensions of finite-dimensional Lie algebras

To make the point explicit, we will again limit our analysis to the example of eleven-dimensional supergravity. Our starting point is then the Lagrangian for this theory compactified on an 8-torus, T8, to D = 2 + 1 spacetime dimensions (after all form fields have been dualized into scalars),
$$\mathcal{L}_{(3)}^{{\rm{SUGR}}{{\rm{A}}_{11}}} = {R_{(3)}} \star {\bf{1}} - \sum\limits_{i = 1}^8 \star d{\hat{\varphi} ^{(i)}} \wedge d{\hat{\varphi} ^{(i)}} - {1 \over 2}\sum\limits_{q = 1}^{120} {{e^{2{\alpha _q}(\hat{\varphi})}}} \star (d{\hat \chi ^{(q)}} + \cdots) \wedge (d{\hat \chi ^{(q)}} + \cdots){.}$$
(5.46)
The second two terms in this Lagrangian correspond to the coset model Ɛ8(8)/(Spin(16)/ℤ2), where Ɛ8(8) denotes the group obtained by exponentiation of the split form E8(8) of the complex Lie algebra E8 and Spin(16)/ℤ2 is the maximal compact subgroup of Ɛ8(8) [33, 134, 35]. The 8 dilatons \(\hat \varphi\) and the 120 axions χ(q) are coordinates on the coset space17. Furthermore, the \({\alpha _q}(\hat \varphi)\) are linear forms on the elements of the Cartan subalgebra \(h = {\hat \varphi ^i}\alpha _i^ \vee\) and they correspond to the positive roots of E8(8) 18. As before, we do not write explicitly the corrections to the curvatures \(d\hat \chi\) that appear in the compactification process. The entire set of positive roots can be obtained by taking linear combinations of the seven simple roots of \(\mathfrak{sl}(8,\,\mathbb{R})\) (we omit the “hatted” notation on the roots since there is no longer any risk of confusion),
$$\begin{array}{*{20}c} {{\alpha _1}(\hat{\varphi}) = {1 \over {\sqrt 2}}\left({{{\sqrt 7} \over 2}{{\hat{\varphi}}_2} - {3 \over 2}{{\hat{\varphi}}_1}} \right),\quad \quad \quad {\alpha _2}(\hat{\varphi}) = {1 \over {\sqrt 2}}\left({{{2\sqrt 3} \over {\sqrt 7}}{{\hat{\varphi}}_3} - {4 \over {\sqrt 7}}{{\hat{\varphi}}_2}} \right),\;\;} \\ {{\alpha _3}(\hat{\varphi}) = {1 \over {\sqrt 2}}\left({{{\sqrt 5} \over {\sqrt 3}}{{\hat{\varphi}}_4} - {{\sqrt 7} \over {\sqrt 3}}{{\hat{\varphi}}_3}} \right),\quad \quad {\alpha _4}(\hat{\varphi}) = {1 \over {\sqrt 2}}\left({{{2\sqrt 2} \over {\sqrt 5}}{{\hat{\varphi}}_5} - {{2\sqrt 3} \over {\sqrt 5}}{{\hat{\varphi}}_4}} \right),\;} \\ {{\alpha _5}(\hat{\varphi}) = {1 \over {\sqrt 2}}\left({{{\sqrt 3} \over {\sqrt 2}}{{\hat{\varphi}}_6} - {{\sqrt 5} \over {\sqrt 2}}{{\hat{\varphi}}_5}} \right),\quad \quad {\alpha _6}(\hat{\varphi}) = {1 \over {\sqrt 2}}\left({{2 \over {\sqrt 3}}{{\hat{\varphi}}_6} - {{2\sqrt 2} \over {\sqrt 3}}{{\hat{\varphi}}_5}} \right),\quad} \\ {{\alpha _7}(\hat{\varphi}) = {1 \over {\sqrt 2}}\left({{{\hat{\varphi}}_8} - \sqrt 3 {{\hat{\varphi}}_7}} \right),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;} \\ \end{array}$$
(5.47)
and the exceptional root
$${\alpha _{10}}(\hat{\varphi}) = {1 \over {\sqrt 2}}\left({{{\hat{\varphi}}_1} + {3 \over {\sqrt 7}}{{\hat{\varphi}}_2} + {{2\sqrt 3} \over {\sqrt 7}}{{\hat{\varphi}}_3}} \right).$$
(5.48)
These correspond exactly to the root vectors \({\vec b_{i,\,i + 1}}\) and \({\vec a_{123}}\) as they appear in the analysis of [35], except for the additional factor of \({1 \over {\sqrt 2}}\) needed to compensate for the fact that the aforementioned reference has an additional factor of 2 in the Killing form. Hence, using the Euclidean metric δ ij (i, j = 1, ⋯, 8) one may check that the roots defined above indeed reproduce the Cartan matrix of E8.
Next, we want to determine the billiard structure for this Lagrangian. As was briefly mentioned before, in the reduction from eleven to three dimensions all the non-gravity walls associated to the eleven-dimensional 3-form A(3) have been transformed, in the same spirit as for the example given above, into electric and magnetic walls of the axionic scalars \(\hat \chi\). Since the terms involving the electric fields \({\partial _t}{\hat \chi ^{(i)}}\) possess no spatial indices, the corresponding wall forms do not contain any of the remaining scale factors \({\hat \beta ^9},\,{\hat \beta ^{10}}\), and are simply linear forms on the dilatons only. In fact the dominant electric wall forms are just the simple roots of E8,
$$\begin{array}{*{20}c} {\hat e_a^{\hat \chi}(\hat{\varphi}) = {\alpha _a}(\hat{\varphi})\qquad (a = 1, \cdots ,7),} \\ {\hat e_{10}^{\hat \chi}(\hat{\varphi}) = {\alpha _{10}}(\hat{\varphi})\quad \quad \quad \quad \quad \quad \;\;} \\ \end{array}$$
(5.49)
The magnetic wall forms naturally come with one factor of \(\hat \beta\) since the magnetic field strength \({\partial _i}\hat \chi\) carries one spatial index. The dominant magnetic wall form is then given by
$$\hat m_9^{\hat \chi}(\hat \beta ,\hat{\varphi}) = {\hat \beta ^9} - \theta (\hat{\varphi}),$$
(5.50)
where \(\theta (\hat \varphi)\) denotes the highest root of E8 which takes the following form in terms of the simple roots,
$$\theta = 2{\alpha _1} + 4{\alpha _2} + 6{\alpha _3} + 5{\alpha _4} + 4{\alpha _5} + 3{\alpha _6} + 2{\alpha _7} + 3{\alpha _{10}} = \sqrt 2 \,{\hat{\varphi} _8}.$$
(5.51)
Since we are in three dimensions there is no curvature wall and hence the only wall associated to the Einstein-Hilbert term is the symmetry wall
$${\hat s_9} = {\hat \beta ^{10}} - {\hat \beta ^9},$$
(5.52)
coming from the three-dimensional metric ĝ μv (μ, v = 0, 9, 10). We have thus found all the dominant wall forms in terms of the lower-dimensional variables.

The structure of the corresponding Lorentzian Kac-Moody algebra is now easy to establish in view of our discussion of overextensions in Section 4.9. The relevant walls listed above are the simple roots of the (untwisted) overextension \(E_8^{+ +}\). Indeed, the relevant electric roots are the simple roots of E8, the magnetic root of Equation (5.50) provides the affine extension, while the gravitational root of Equation (5.52) is the overextended root.

What we have found here in the case of eleven-dimensional supergravity also holds for the other theories with U-duality algebra \(\mathfrak{u}_3\) in 3 dimensions when \(\mathfrak{u}_3\) is a split real form. The Coxeter group and the corresponding Kac-Moody algebra are given by the untwisted overextension \(\mathfrak{u}_3^{++}\). This overextension emerges as follows [41]:
  • The dominant electric wall forms \({\hat e^{\hat \chi}}(\hat \varphi)\) for the supergravity theory in question are in one-to-one correspondence with the simple roots of the associated U-duality algebra \(\mathfrak{u}_3\).

  • Adding the dominant magnetic wall form \({\hat m^{\hat \chi}}(\hat \beta, \,\hat \varphi) = {\hat \beta ^9} - \theta (\hat \varphi)\) corresponds to an affine extension \(\mathfrak{u}_3^{+}\) of \(\mathfrak{u}_3\).

  • Finally, completing the set of dominant wall forms with the symmetry wall \({\hat s_9}(\hat \beta) = {\hat \beta ^{10}} - {\hat \beta ^9}\), which is the only gravitational wall form existing in three dimensions, is equivalent to an overextension \(\mathfrak{u}_3^{++}\) of \(\mathfrak{u}_3\).

Thus we see that the appearance of overextended algebras in the BKL-analysis of supergravity theories is a generic phenomenon closely linked to hidden symmetries.

5.4 Models associated with split real forms

In this section we give a complete list of all theories whose billiard description can be given in terms of a Kac-Moody algebra that is the untwisted overextension of a split real form of the associated U-duality algebra (see Table 15). These are precisely the maximally oxidized theories introduced in [22] and further examined in [37]. These theories are completely classified by their global symmetry groups \({{\mathcal U}_3}\) arising in three dimensions. For the string-related theories the group \({{\mathcal U}_3}\) is the (classical version of) the U-duality symmetry obtained by combining the S- and T-dualities in three dimensions [142]. Thereof the notation \({{\mathcal U}_3}\) for the global symmetry group in three dimensions. We extend the classification to the non-split case in Section 7.
Table 15

We present here the complete list of theories that exhibit extended coset symmetries of split real Lie algebras upon compactification to three spacetime dimensions. In the leftmost column we give the coset space which is relevant in each case. We also list the Kac-Moody algebras that underlie the gravitational dynamics in the BKL-limit. These appear as overextensions of the finite Lie algebras found in three dimensions. Finally we indicate which of these theories are related to string/M-theory.

\(\mathcal{U}_3/\mathcal{K}(\mathcal{U}_3)\)

Lagrangian in maximal dimension

Kac-Moody algebra

String/M-theory

\({{SL(n + 1,{\mathbb R})} \over {SO(n + 1)}}\)

\({\mathcal{L}}_{n+3} = R\star \bf{1}\)

\(AE_{n+2}\equiv A_n^{++}\)

No

\({{SO(n,n + 1)} \over {SO(n) \times SO(n + 1)}}\)

\(\begin{array}{*{20}c} {{{\mathcal L}_{n + 2}} = R\star {\bf 1} - \star d\phi \wedge d\phi - {1 \over 2}{e^{2{{\sqrt 2} \over {\sqrt n}}\phi}}\star {G^{(3)}}\wedge{G^{(3)}} - {1 \over 2}{e^{{2 \over {\sqrt n}}\phi}}\star {F^{(2)}}\wedge{F^{(2)}},} \\ {{G^{(3)}} = d{B^{(2)}} + {1 \over 2}{A^{(1)}}{\wedge^{(1)}},\quad \quad {F^{(2)}} = d{A^{(1)}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}\)

\(B{E_{n + 2}} \equiv B_n^{+ +}\)

No

\({{Sp(n)} \over {U(n)}}\)

\(\begin{array}{*{20}c} {{{\mathcal L}_4} = R\star {\bf{1}} - \star d\vec \phi \wedge d\vec \phi - {1 \over 2}\sum\nolimits_\alpha {{e^{2{{\vec \sigma}_{\alpha \cdot}}\vec \phi}}\star (d{{\mathcal X}^\alpha} + \cdots)\wedge (d{{\mathcal X}^\alpha} + \cdots) -}} \\ {{1 \over 2}\sum\nolimits_{a = 1}^{n - 1} {{e^{{{\vec e}_{a \cdot}}\vec \phi \sqrt 2}}\star dA_{(1)}^a\wedge dA_{(1)}^a} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}\)

\(C{E_{n + 2}} \equiv C_n^{+ +}\)

No

\({{SO(n,n)} \over {SO(n) \times SO(n)}}\)

\({{\mathcal L}_{n + 2}} = R\star {\bf{1}} - \star d\phi \wedge d\phi - {1 \over 2}{e^{{4 \over {\sqrt n}}\phi}}\star d{B^{(2)}}\wedge d{B^{(2)}}\)

\(D{E_{n + 2}} \equiv D_n^{+ +}\)

type I (n = 8) / bosonic string (n = 24)

\({{{G_{2(2)}}} \over {SU(8)}}\)

\({{\mathcal L}_5} = R\star {\bf{1}} - {1 \over 2}\star {F^{(2)}}\wedge {F^{(2)}} + \,\,{1 \over {3\sqrt 3}}{F^{(2)}}\wedge {F^{(2)}}\wedge {A^{(1)}},\,\,{F^{(2)}} = d{A^{(1)}}\)

\(G_2^{+ +}\)

No

\({{{F_{4(4)}}} \over {Sp(3) \times SU(3)}}\)

\(\begin{array}{*{20}{c}} {{\mathcal{L}_6} = R \star {\mathbf{1}} - \star d\phi \wedge d\phi - \frac{1}{2}{e^{2\phi }} \star d\chi \wedge d\chi - \frac{1}{2}{e^{ - 2\phi }} \star {H^{(3)}} \wedge {H^{(3)}} - \frac{1}{2} \star {G^{(3)}} \wedge \quad } \\ {{G^{(3)}} - \frac{1}{2}{e^\phi } \star F_{(2)}^ + \wedge F_{(2)}^ + - \frac{1}{2}{e^{ - \phi }} \star F_{(2)}^ - \wedge F_{(2)}^ - - \frac{1}{{\sqrt 2 }}\chi {H^{(3)}} \wedge {G^{(3)}} - {\mkern 1mu} {\mkern 1mu} \quad \quad \quad \quad \quad } \\ {\frac{1}{2}A_{(1)}^{{\kern 1pt} + } \wedge F_{(2)}^ + \wedge {H^{(3)}} - \frac{1}{2}A_{(1)}^{{\kern 1pt} + } \wedge F_{(2)}^ - \wedge {G^{(3)}},\quad \quad F_{(2)}^ + = dA_{(1)}^{{\kern 1pt} + } + {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \frac{1}{{\sqrt 2 }}\chi dA_{(1)}^{{\kern 1pt} - },\quad \quad } \\ {F_{(2)}^ - = dA_{(1)}^{{\kern 1pt} - },\quad \quad {H^{(3)}} = d{B^{(2)}} + \frac{1}{2}A_{(1)}^{{\kern 1pt} - } \wedge dA_{(1)}^{{\kern 1pt} - },\quad \quad {G^{(3)}} = d{C^{(2)}} - \quad \quad \quad \quad \quad {\mkern 1mu} {\mkern 1mu} } \\ {\frac{1}{{\sqrt 2 }}\chi {H^{(3)}} - \frac{1}{2}A_{(1)}^{{\kern 1pt} + } \wedge dA_{(1)}^{{\kern 1pt} - }{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \end{array}\)

\(F_4^{+ +}\)

No

\({{{E_{6(6)}}} \over {Sp(4)/{{\mathbb Z}_2}}}\)

\(\begin{array}{*{20}c} {{{\mathcal L}_8} = R\star {\bf{1}} - \star d\phi \wedge d\phi - {1 \over 2}{e^{2\sqrt 2 \phi}}\star d{\chi}\wedge d{\chi} - {1 \over 2}{e^{- \sqrt 2 \phi}}\star {G^{(4)}}\wedge {G^{(4)}} +} \\ {{\chi}{G^{(4)}}\wedge {G^{(4)}},\quad \quad {G^{(4)}} = d{C^{(3)}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}\)

\(E_6^{+ +}\)

No

\({{{E_{7(7)}}} \over {SU(8)/{{\mathbb Z}_2}}}\)

\(\begin{array}{*{20}{c}} {{L_9} = R \star 1 - \; \star d\phi \wedge d\phi - \tfrac{1}{2}e\tfrac{{2\sqrt 2 }}{{\sqrt 7 }}\phi \star d{C^{(3)}} \wedge d{C^{(3)}} - } \\ {\tfrac{1}{2}e - \tfrac{{4\sqrt 2 }}{{\sqrt 7 }}\phi \star d{A^{(1)}} \wedge d{A^{(1)}} - \tfrac{1}{2}d{C^{(3)}} \wedge {A^{(1)}}\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}\)

\(E_7^{+ +}\)

No

\({{{E_{8(8)}}} \over {{\rm{Spin(16)/}}{{\mathbb Z}_2}}}\)

\({{\mathcal L}_{11}} = R \star {\bf{1}} - {1 \over 2} \star d{C^{(3)}} \wedge d{C^{(3)}} - {1 \over 6}d{C^{(3)}} \wedge d{C^{(3)}} \wedge {C^{(3)}}\)

\({E_{10}} = E_8^{+ +}\)

M-theory, type IIA and type IIB string theory

Let us also note here that, as shown in [55], the billiard analysis sheds light on the problem of oxidation, i.e., the problem of finding the maximum spacetime dimension in which a theory with a given duality group in three dimensions can be reformulated. More on this question can be found in [118, 119].

6 Finite-Dimensional Real Lie Algebras

In this section we explain the basic theory of real forms of finite-dimensional Lie algebras. This material is somewhat technical and may therefore be skipped at a first reading. The theory of real forms of Lie algebras is required for a complete understanding of Section 7, which deals with the general case of Kac-Moody billiards for non-split real forms. However, for the benefit of the reader who wishes to proceed directly to the physical applications, we present a brief summary of the main points in the beginning of Section 7.

Our intention with the following presentation is to provide an accessible reference on the subject, directed towards physicists. We therefore consider this section to be somewhat of an entity of its own, which can be read independently of the rest of the paper. Consequently, we introduce Lie algebras in a rather different manner compared to the presentation of Kac-Moody algebras in Section 4, emphasizing here more involved features of the general structure theory of real Lie algebras rather than relying entirely on the Chevalley-Serre basis and its properties. In the subsequent section, the reader will then see these two approaches merged, and used simultaneously to describe the billiard structure of theories whose U-duality algebras in three dimensions are given by arbitrary real forms.

We have adopted a rather detailed and explicit presentation. We do not provide all proofs, however, referring the reader to [93, 129, 133, 94] for more information (including definitions of basic Lie algebra theory concepts).

There are two main approaches to the classification of real forms of finite-dimensional Lie algebras. One focuses on the maximal compact Cartan subalgebra and leads to Vogan diagrams. The other focuses on the maximal noncompact Cartan subalgebra and leads to Tits-Satake diagrams. It is this second approach that is of direct use in the billiard analysis. However, we have chosen to present here both approaches as they mutually enlighten each other.

6.1 Definitions

Lie algebras are usually, in a first step at least, considered as complex, i.e., as complex vector spaces, structured by an antisymmetric internal bilinear product, the Lie bracket, obeying the Jacobi identity. If {T α } denotes a basis of such a complex Lie algebra \({\mathfrak g}\) of dimension n (over ℂ), we may also consider \({\mathfrak g}\) as a real vector space of double dimension 2 n (over ℝ), a basis being given by {T α , iT α }. Conversely, if \({{\mathfrak g}_0}\) is a real Lie algebra, by extending the field of scalars from ℝ to ℂ, we obtain the complexification of \({{\mathfrak g}_0}\), denoted by \({{\mathfrak g}^{\mathbb C}}\), defined as:
$${{\mathfrak g}^{\mathbb{C}}} = {{\mathfrak g}_0}{\otimes _{\mathbb R}}\mathbb C.$$
(6.1)
Note that \({({{\mathfrak g}^{\mathbb C}})^{\mathbb R}} = {{\mathfrak g}_0} \oplus i{{\mathfrak g}_0}\) and \({\dim _{\mathbb R}}{({{\mathfrak g}^{\mathbb C}})^{\mathbb R}} = 2\,{\dim _{\mathbb R}}({{\mathfrak g}_0})\). When a complex Lie algebra \({\mathfrak g}\), considered as a real algebra, has a decomposition
$${{\mathfrak g}^{\mathbb R}} = {{\mathfrak g}_0} \oplus i{{\mathfrak g}_0},$$
(6.2)
with \({{\mathfrak g}_0}\) being a real Lie algebra, we say that \({{\mathfrak g}_0}\) is a real form of the complex Lie algebra \({\mathfrak g}\). In other words, a real form of a complex algebra exists if and only if we may choose a basis of the complex algebra such that all the structure constants become real. Note that while \({\mathfrak g}^{\mathbb R}\) is a real space, multiplication by a complex number is well defined on it since \({{\mathfrak g}_0} \oplus i{{\mathfrak g}_0} = {{\mathfrak g}_0}{\otimes _{\mathbb R}}{\mathbb C}\). As we easily see from Equation (6.2),
$${\mathbb C} \times {{\mathfrak g}^\mathbb R}\quad \rightarrow \quad {{\mathfrak g}^\mathbb R}:(a + i b,{X_0} + i {Y_0}) \mapsto (a{X_0} - b{Y_0}) + i (a{Y_0} + b{X_0}),,$$
(6.3)
where a, b ∈ ℝ and X0, Y0\({Y_0} \in {{\mathfrak g}_0}\).
The Killing form is defined by
$$B(X,Y) = {\rm{Tr}}({\rm{ad}}\,X\,{\rm{ad}}\,Y)$$
(6.4)
The Killing forms on \({\mathfrak g}^{\mathbb R}\) and \({\mathfrak g}^{\mathbb C}\) or \({{\mathfrak g}_0}\) are related as follows. If we split an arbitrary generator Z of \({\mathfrak g}\) according to Equation (6.2) as Z = X0 + iY0, we may write:
$${B_{{{\mathfrak g}^\mathbb R}}}(Z, Z\prime) = 2{{\rm Re}}\, {B_{{{\mathfrak g}^\mathbb C}}}(Z, Z\prime) = 2({B_{{{\mathfrak g}_0}}}({X_0}, {X\prime _0}) - {B_{{{\mathfrak g}_0}}}({Y_0}, \,{Y\prime _0})).$$
(6.5)
Indeed, if \({\rm{a}}{{\rm{d}}_{\mathfrak g}}Z\) Z is a complex n × n matrix, \({\rm{a}}{{\rm{d}}_{{{\mathfrak g}^{\mathbb R}}}}({X_0} + i{Y_0})\) is a real 2n × 2n matrix:
$$a{d_{{{\mathfrak g}^\mathbb R}}}({X_0} + i {Y_0}) = \left(\begin{array}{*{20}c}{{\rm{a}}{{\rm{d}}_{{{\mathfrak g}_0}}}{X_0}} & {- {\rm{a}}{{\rm{d}}_{{{\mathfrak g}_0}}}{Y_0}} \\{{\rm{a}}{{\rm{d}}_{{{\mathfrak g}_0}}}{Y_0}} & {{\rm{a}}{{\rm{d}}_{{{\mathfrak g}_0}}}{X_0}} \\ \end{array} \right).$$
(6.6)

6.2 A preliminary example: \({\mathfrak {sl}}(2,\,{\mathbb C})\)

Before we proceed to develop the general theory of real forms, we shall in this section discuss in detail some properties of the real forms of \({A_1} = {\mathfrak {sl}}(2,\,{\mathbb C})\). This is a nice example, which exhibits many properties that turn out not to be specific just to the case at hand, but are, in fact, valid also in the general framework of semi-simple Lie algebras. The main purpose of subsequent sections will then be to show how to extend properties that are immediate in the case of \({\mathfrak {sl}}(2,\,{\mathbb C})\), to general semi-simple Lie algebras.

6.2.1 Real forms of \({\mathfrak {sl}}(2,\,{\mathbb C})\)

The complex Lie algebra \({\mathfrak {sl}}(2,\,{\mathbb C})\) can be represented as the space of complex linear combinations of the three matrices
$$h = \left(\begin{array}{*{20}c} 1 & 0 \\ 0 & {- 1} \\ \end{array} \right),\qquad e = \left(\begin{array}{*{20}c} 0 & 1 \\ 0 & 0 \\ \end{array} \right),\qquad f = \left(\begin{array}{*{20}c} 0 & 0 \\ 1 & 0 \\ \end{array} \right)$$
(6.7)
which satisfy the well known commutation relations
$$[h,e] = 2{\mkern 1mu} e,\quad \quad [h,f] = - 2{\mkern 1mu} f,\quad \quad [e,f] = h.$$
(6.8)
A crucial property of these commutation relations is that the structure constants defined by the brackets are all real. Thus by restricting the scalars in the linear combinations from the complex to the real numbers, we still obtain closure for the Lie bracket on real combinations of h, e and f, defining thereby a real form of the complex Lie algebra \({\mathfrak {sl}}(2,\,{\mathbb C})\): the real Lie algebra \({\mathfrak {sl}}(2,\,{\mathbb C})\) 19. As we have indicated above, this real form of \({\mathfrak {sl}}(2,\,{\mathbb C})\) is called the “split real form”.
Another choice of \({\mathfrak {sl}}(2,\,{\mathbb C})\) generators that, similarly, leads to a real Lie algebra consists in taking i times the Pauli matrices σ x , σ y , σ z , i.e.,
$${\tau ^x} = i(e + f) = \left(\begin{array}{*{20}c} 0 & i \\ i & 0 \\ \end{array} \right),\quad \quad {\tau ^y} = (e - f) = \left(\begin{array}{*{20}c} 0 & 1 \\ {- 1} & 0 \\ \end{array} \right),\quad \quad {\tau ^z} = ih = \left(\begin{array}{*{20}c} i & 0 \\ 0 & {- i} \\ \end{array} \right).$$
(6.9)
The real linear combinations of these matrices form the familiar \({\mathfrak {su}}(2)\) Lie algebra (a real Lie algebra, even if some of the matrices using to represent it are complex). This real Lie algebra is non-isomorphic (as a real algebra) to \({\mathfrak {sl}}(2,\,{\mathbb R})\) as there is no real change of basis that maps {h, e, f} on a basis with the \({\mathfrak {su}}(2)\) commutation relations. Of course, the two algebras are isomorphic over the complex numbers.

6.2.2 Cartan subalgebras

Let \({\mathfrak h}\) be a subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\). We say that \({\mathfrak h}\) is a Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\) if it is a Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb C})\) when the real numbers are replaced by the complex numbers. Two Cartan subalgebras \({\mathfrak {h}_1}\) and \({\mathfrak {h}_2}\) of \({\mathfrak {sl}}(2,\,{\mathbb R})\) are said to be equivalent (as Cartan subalgebras of \({\mathfrak {sl}}(2,\,{\mathbb R})\) if there is an automorphism a of \({\mathfrak {sl}}(2,\,{\mathbb R})\) such that \(a({\mathfrak {h}_1}) = {\mathfrak {h}_2}\).

The subspace ℝh constitutes clearly a Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\). The adjoint action of h is diagonal in the basis {e, f, h} and can be represented by the matrix
$$\left(\begin{array}{*{20}c}2 & {\quad 0} & 0 \\ 0 & {\,\, - 2} & 0 \\ 0 & {\quad 0} & 0 \\ \end{array} \right).$$
(6.10)
Another Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\) is given by ℝ(ef) ≡ ℝτ y , whose adjoint action with respect to the same basis is represented by the matrix
$$\left(\begin{array}{*{20}c}{\,\,\,\,0} & 1 & 1 \\ {- 2} & 0 & 0 \\ {- 2} & 0 & 0 \\ \end{array} \right).$$
(6.11)
Contrary to the matrix representing ad h , in addition to 0 this matrix has two imaginary eigenvalues: ±2i. Thus, there can be no automorphism a of \({\mathfrak {sl}}(2,\,{\mathbb R})\) such that τ y = λa(h), λ ∈ ℝ since ada(h) has the same eigenvalues as ad h , implying that the eigenvalues of λ ada(h) are necessarily real (λ ∈ ℝ).
Consequently, even though they are equivalent over the complex numbers since there is an automorphism in SL(2, ℂ) that connects the complex Cartan subalgebras ℂ h and ℂ τ y , we obtain
$${\tau ^y} = i{\rm{Ad}}\left({{\rm{Exp}}\left[ {i{\pi \over 4}(e + f)} \right]} \right)h,\quad \quad h = i \;{\rm{Ad}}\left({{\rm{Exp}}\left[ {{\pi \over 4}{\tau ^x}} \right]} \right){\tau ^y}.$$
(6.12)
The real Cartan subalgebras generated by h and τ y are non-isomorphic over the real numbers.

6.2.3 The Killing form

The Killing form of SL(2, ℝ) reads explicitly
$$B = \left(\begin{array}{*{20}c}0 & 4 & 0 \\ 4 & 0 & 0 \\ 0 & 0 & 8 \\ \end{array} \right)$$
(6.13)
in the basis {e, f, h}. The Cartan subalgebra ℝh is spacelike while the Cartan subalgebra ℝτ y is timelike. This is another way to see that these are inequivalent since the automorphisms of \({\mathfrak {sl}}(2,\,{\mathbb R})\) preserve the Killing form. The group \({\rm{Aut}[\mathfrak {sl}}(2,\,{\mathbb R})]\) of automorphisms of \({\mathfrak {sl}}(2,\,{\mathbb R})\) is SO(2, 1), while the subgroup Int[\({\rm{Int}[\mathfrak {sl}}(2,\,{\mathbb R})] \subset {\rm{Aut}[\mathfrak {sl}}(2,\,{\mathbb R})]\) of inner automorphisms is the connected component SO(2, 1)+ of SO(2, 1). All spacelike directions are equivalent, as are all timelike directions, which shows that all the Cartan subalgebras of \({\mathfrak {sl}}(2,\,{\mathbb R})\) can be obtained by acting on these two inequivalent particular ones by \({\rm{Int}[\mathfrak {sl}}(2,\,{\mathbb R})]\), i.e., the adjoint action of the group SL(2, ℝ). The lightlike directions do not define Cartan subalgebras because the adjoint action of a lighlike vector is non-diagonalizable. In particular ℝe and ℝf are not Cartan subalgebras even though they are Abelian.
By exponentiation of the generators h and τ y , we obtain two subgroups, denoted \({\mathcal A}\) and \({\mathcal K}\):
$$\mathcal A = \left\{{{\rm Exp} [t\;h] = \left(\begin{array}{*{20}c}{{e^t}} & 0 \\ {\,0} & {{e^{- t}}} \\ \end{array} \right)\vert t \in {\mathbb R}} \right\} \simeq \mathbb R,$$
(6.14)
$$\mathcal{K} = \left\{{{\rm{Exp}}[t\,{\tau ^y}] = \left({\begin{array}{*{20}c}{\cos (t)} & {\sin (t)}\\{- \sin (t)} & {\cos (t)} \end{array}} \right)\vert t \in \left[ {0,2\pi} \right[} \right\}\, \simeq {\mathbb{R}}/{\mathbb{Z}}{.}$$
(6.15)
The subgroup defined by Equation (6.14) is noncompact, the one defined by Equation (6.15) is compact; consequently the generator h is also said to be noncompact while τ y is called compact.

6.2.4 The compact real form \({\mathfrak {su}}(2)\)

The Killing metric on the group \({\mathfrak {su}}(2)\) is negative definite. In the basis {τ x , τ y , τ z }, it reads
$$B = \left({\begin{array}{*{20}c}{- 8} & 0 & 0\\0 & {- 8} & 0\\0 & 0 & {- 8}\end{array}} \right){.}$$
(6.16)

The corresponding group obtained by exponentiation is SU(2), which is isomorphic to the 3-sphere and which is accordingly compact. All directions in \({\mathfrak {su}}(2)\) are equivalent and hence, all Cartan subalgebras are SU(2) conjugate to ℝτ y . Any generator provides by exponentiation a group isomorphic to ℝ/ℤ and is thus compact.

Accordingly, while \({\mathfrak {sl}}(2,\,{\mathbb R})\) admits both compact and noncompact Cartan subalgebras, the Cartan subalgebras of \({\mathfrak {su}}(2)\) are all compact. The real algebra \({\mathfrak {su}}(2)\) is called the compact real form of \({\mathfrak {sl}}(2,\,{\mathbb C})\). One often denotes the real forms by their signature. Adopting Cartan’s notation A1 for \({\mathfrak {sl}}(2,\,{\mathbb C})\), one has \({\mathfrak {sl}}(2,\,{\mathbb R}) \equiv {A_{1(1)}}\) and \({\mathfrak {su}}(2) \equiv {A_{1(-3)}}\). We shall verify before that there are no other real forms of \({\mathfrak {sl}}(2,\,{\mathbb C})\).

6.2.5 \({\mathfrak {su}}(2)\) and \({\mathfrak {sl}}(2,\,{\mathbb R})\) compared and contrasted — The Cartan involution

Within \({\mathfrak {sl}}(2,\,{\mathbb C})\), one may express the basis vectors of one of the real subalgebras \({\mathfrak {su}}(2)\) or \({\mathfrak {sl}}(2,\,{\mathbb R})\) in terms of those of the other. We obtain, using the notations t = (ef) and x = (e + f):
$$\begin{array}{*{20}c} {x = - i\,{\tau ^x},} & {{\tau ^x} = i\,x,}\\ {h = - i\,{\tau ^z},} & {{\tau ^z} = i\,h,}\\ {t = {\tau ^y},\;\;\;} & {{\tau ^y} = t{.}\;\;\;} \end{array}$$
(6.17)
Let us remark that, in terms of the generators of \({\mathfrak {su}}(2)\), the noncompact generators x and h of \({\mathfrak {sl}}(2,\,{\mathbb R})\) are purely imaginary but the compact one t is real.
More precisely, if τ denotes the conjugation20 of \({\mathfrak {sl}}(2,\,{\mathbb C})\) that fixes {τ x , τ y , τ z }, we obtain:
$$\tau (x) = - x,\qquad \tau (t) = + t,\qquad \tau (h) = - h,$$
(6.18)
or, more generally,
$$\forall X \in \mathfrak{sl}(2,{\mathbb{C}}):\tau (X) = - {X^\dagger}.$$
(6.19)
Conversely, if we denote by a the conjugation of \({\mathfrak {sl}}(2,\,{\mathbb C})\) that fixes the previous \({\mathfrak {sl}}(2,\,{\mathbb R})\) Cartan subalgebra in \({\mathfrak {sl}}(2,\,{\mathbb C})\), we obtain the usual complex conjugation of the matrices:
$$\sigma (X) = \,\overline {X} .$$
(6.20)

The two conjugations τ and a of \({\mathfrak {sl}}(2,\,{\mathbb C})\) associated with the real subalgebras \({\mathfrak {su}}(2)\) and \({\mathfrak {sl}}(2,\,{\mathbb R})\) of \({\mathfrak {sl}}(2,\,{\mathbb C})\) commute with each other. Each of them, trivially, fixes pointwise the algebra defining it and globally the other algebra, where it constitutes an involutive automorphism (“involution”).

The Killing form is neither positive definite nor negative definite on \({\mathfrak {sl}}(2,\,{\mathbb R})\): The symmetric matrices have positive norm squared, while the antisymmetric ones have negative norm squared. Thus, by changing the relative sign of the contributions associated with symmetric and antisymmetric matrices, one can obtain a bilinear form which is definite. Explicitly, the involution θ of \({\mathfrak {sl}}(2,\,{\mathbb R})\) defined by θ(X) = −X t has the feature that
$${B^\theta}(X,Y) = - B(X,\theta Y)$$
(6.21)
is positive definite. An involution of a real Lie algebra with that property is called a “Cartan involution” (see Section 6.4.3 for the general definition).

The Cartan involution θ is just the restriction to \({\mathfrak {sl}}(2,\,{\mathbb R})\) of the conjugation τ associated with the compact real form \({\mathfrak {su}}(2)\) since for real matrices X = X t . One says for that reason that the compact real form \({\mathfrak {su}}(2)\) and the noncompact real form \({\mathfrak {sl}}(2,\,{\mathbb R})\) are “aligned”.

Using the Cartan involution θ, one can split \({\mathfrak {sl}}(2,\,{\mathbb R})\) as the direct sum
$$\mathfrak{sl}(2,\mathbb{R}) = {\mathfrak{t}} \oplus {\mathfrak{p}},$$
(6.22)
where \({\mathfrak k}\) is the subspace of antisymmetric matrices corresponding to the eigenvalue +1 of the Cartan involution while \({\mathfrak p}\) is the subspace of symmetric matrices corresponding to the eigenvalue −1. These are also eigenspaces of τ and given explicitly by \({\mathfrak k}={\mathbb R}t\) and \({\mathfrak p} = {\mathbb R}x \oplus {\mathbb R}h\). One has
$$\mathfrak{su}(2) =\mathfrak{t}\oplus i\mathfrak{p},$$
(6.23)
i.e., the real form \({\mathfrak {sl}}(2,\,{\mathbb R})\) is obtained from the compact form \({\mathfrak {su}}(2)\) by inserting an “i” in front of the generators in \({\mathfrak p}\).

6.2.6 Concluding remarks

Let us close these preliminaries with some remarks.
  1. 1.
    The conjugation τ allows to define a Hermitian form on \({\mathfrak {sl}}(2,\,{\mathbb C})\):
    $$X \bullet Y = - {\rm{Tr}}(Y\tau (X)).$$
    (6.24)
     
  2. 2.
    Any element of the group SL(2, ℝ) can be written as a product of elements belonging to the subgroups \({\mathcal K}\), \({\mathcal A}\) and \({\mathcal N} = {\rm{Exp}}[{\mathbb R}e]\) (Iwasawa decomposition),
    $${\rm{Exp}}[\theta \,t]\;{\rm{Exp}}[a\,h]\;{\rm{Exp}}[n\,e] = \left({\begin{array}{*{20}c} {{e^a}\cos \theta} & {n\,{e^a}\cos \theta + {e^{- a}}\sigma n\theta}\\ {- {e^a}\sigma n\theta} & {{e^{- a}}\cos \theta - n\,{e^a}\sigma n\theta} \end{array}} \right){.}$$
    (6.25)
     
  3. 3.
    Any element of \({\mathfrak p}\) is conjugated via \({\mathcal K}\) to a multiple of h,
    $$\rho (\cos \alpha h + \sin\alpha x) = \left({\begin{array}{*{20}c} {\cos {{\alpha} \over 2}} & {\sin{{\alpha} \over 2}}\\ {- \sin{{\alpha} \over 2}} & {\cos {{\alpha} \over 2}} \end{array}} \right)\rho \,h\;\left({\begin{array}{*{20}c} {\cos {{\alpha} \over 2}} & {- \sin{{\alpha} \over 2}}\\ {\sin{{\alpha} \over 2}} & {\cos {{\alpha} \over 2}} \end{array}} \right),$$
    (6.26)
    so, denoting by \({\mathfrak a}={\mathbb R}h\) the (maximal) noncompact Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\), we obtain
    $${\mathfrak{p}} = {\rm{Ad}}({\mathcal K}){\mathfrak{a}}{.}$$
    (6.27)
     
  4. 4.

    Any element of SL(2, ℝ) can be written as the product of an element of \({\mathcal K}\) and an element of Exp[\({\rm{EXP[{\mathfrak p}]}}\)]. Thus, as a consequence of the previous remark, we have \(SL(2,\,{\mathbb R}) = {\mathcal {KAK}}\) (Cartan)21.

     
  5. 5.
    When the Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\) is chosen to be ℝ h, the root vectors are e and f. We obtain the compact element t, generating a non-equivalent Cartan subalgebra by taking the combination
    $$t = e + \theta (e).$$
    (6.28)
    Similarly, the normalized root vectors associated with t are (up to a complex phase) E±2i = ½(hix):
    $${[}t,\;{E_{2i}}] = 2i\,{E_{2i}},\qquad {[}t,\;{E_{- 2i}}] = - 2i\;{E_{- 2i}},\qquad {[}{E_{2i}},\;{E_{- 2i}}] = i\;t.$$
    (6.29)
    Note that both the real and imaginary components of E±2i are noncompact. They allow to obtain the noncompact Cartan generators h, x by taking the combinations
    $$\cos \alpha \; h + \sin\;\alpha \,x = {e^{i\alpha}}{E_{2i}} + {e^{- i\alpha}}{E_{- 2i}}.$$
    (6.30)
     

6.3 The compact and split real forms of a semi-simple Lie algebra

We shall consider here only semi-simple Lie algebras. Over the complex numbers, Cartan sub-algebras are “unique”22. These subalgebras may be defined as maximal Abelian subalgebras \({\mathfrak h}\) such that the transformations in ad[\({\rm{ad[{\mathfrak h}]}}\)] are simultaneously diagonalizable (over ℂ). Diagonalizability is an essential ingredient in the definition. There might indeed exist Abelian subalgebras of dimension higher than the rank (= dimension of Cartan subalgebras), but these would involve non-diagonalizable elements and would not be Cartan subalgebras23.

We denote the set of nonzero roots as Δ. One may complete the Chevalley generators into a full basis, the so-called Cartan-Weyl basis, such that the following commutation relations hold:
$$\left[ {H,\;{E_\alpha}} \right] = \alpha (H)\,{E_\alpha},$$
(6.31)
$$\left[ {{E_\alpha},{E_\beta}} \right] = \left\{{\begin{array}{*{20}c} {{N_{\alpha ,\,\beta}}{E_{\alpha + \beta}}} & {{\rm{if}}\alpha + \beta \in \Delta ,}\\ {{H_\alpha}\quad \quad \;\;} & {{\rm{if}}\alpha + \beta = 0,}\\ {0\quad \quad \quad} & {{\rm{if}}\alpha + \beta \not \in \Delta ,} \end{array}} \right.$$
(6.32)
where H α is defined by duality thanks to the Killing form B(X, Y) = Tr(ad X ad Y), which is non-singular on semi-simple Lie algebras:
$$\forall H \in {\mathfrak{h}}:\alpha (H) = B({H_\alpha},H),$$
(6.33)
and the generators are normalized according to (see Equation (6.43))
$$B({E_\alpha},\,{E_\beta}) = {\delta _{\alpha + \beta ,0}}.$$
(6.34)
The generators E α associated with the roots α (where α need not be a simple root) may be chosen such that the structure constants N α,β satisfy the relations
$${N_{\alpha ,\beta}} = - {N_{\beta ,\alpha}} = - {N_{- \alpha , - \beta}} = {N_{\beta , - \alpha - \beta}},$$
(6.35)
$$N_{\alpha ,\,\beta}^2 = {1 \over 2}q(p + 1)(\alpha \vert \alpha),\qquad p,\,q \in {\mathbb{N}}_0,$$
(6.36)
where the scalar product between roots is defined as
$$(\alpha \vert \beta) = B({H_\alpha},{H_\beta})$$
(6.37)
The non-negative integers p and q are such that the string of all vectors β + n α belongs to Δ for −pnq; they also satisfy the equation pq = 2(β|α)/(α|α). A standard result states that for semi-simple Lie algebras
$$(\alpha \vert \beta) = \sum\limits_{\gamma \in \Delta} {(\alpha \vert} \gamma)(\gamma \vert \beta) \in \mathbb{Q},$$
(6.38)
from which we notice that the roots are real when evaluated on an H β -generator,
$$\alpha ({H_\beta}) = (\alpha \vert \beta).$$
(6.39)
An important consequence of this discussion is that in Equation (6.32), the structure constants of the commutations relations may all be chosen real. Thus, if we restrict ourselves to real scalars we obtain a real Lie algebra \({{\mathfrak s}_0}\), which is called the split real form because it contains the maximal number of noncompact generators. This real form of \({\mathfrak g}\) reads explicitly
$$\mathfrak{s}_0 =\underset{\alpha \in \Delta}{\bigoplus} \mathbb{R}{H_\alpha} \oplus \underset{\alpha \in \Delta}{\bigoplus} \mathbb{R}{E_\alpha}.$$
(6.40)
The signature of the Killing form on \({{\mathfrak s}_0}\) (which is real) is easily computed. First, it is positive definite on the real linear span \({{\mathfrak h}_0}\) of the H α generators. Indeed,
$$B({H_\alpha},\,{H_\alpha}) = (\alpha \vert \alpha) = \sum\limits_{\gamma \in \Delta} {(\alpha \vert} \gamma {)^2} > 0.$$
(6.41)
Second, the invariance of the Killing form fixes the normalization of the E α generators to one,
$$B({E_\alpha},{E_{- \alpha}}) = 1,$$
(6.42)
since24
$$B([{E_\alpha}{E_{- \alpha}}],{H_\alpha}) = (\alpha \vert \alpha) = - B({E_{- \alpha}},[{E_\alpha}{H_\alpha}]) = (\alpha \vert \alpha)B({E_\alpha},{E_{- \alpha}}).$$
(6.43)
Moreover, one has B(g α , g β ) = 0 if α + β ≠ 0. Indeed ad[g α ] ad[g β ] maps g μ into gμ+α+β, i.e., in matrix terms ad[g α ] ad[g β ] has zero elements on the diagonal when α + β ≠ 0. Hence, the vectors E α + Eα are spacelike and orthogonal to the vectors E α Eα, which are timelike. This implies that the signature of the Killing form is
$$\left(\left.\frac{1}{2} (\dim \mathfrak{s}_{0}+{\rm{rank}}\; \mathfrak{s}_{0})\right\vert_{+}, \left.\frac{1}{2} (\dim \mathfrak{s}_{0}-{\rm{rank}}\;\mathfrak{s}_{0})\right\vert_{-}\right){.}$$
(6.44)
The split real form \({{\mathfrak s}_0}\) of \({\mathfrak g}\) is “unique”.
On the other hand, it is not difficult to check that the linear span
$$\mathfrak {c}_{0}=\underset{\alpha \in \Delta}{\bigoplus}{\mathbb R}(i\, H_{\alpha})\oplus \underset{\alpha \in \Delta}{\bigoplus}{\mathbb R}({E}_{\alpha}-{E}_{-\alpha})\oplus \underset{\alpha \in \Delta}{\bigoplus}{\mathbb R}\,i\,({E}_{\alpha}+{E}_{-\alpha})$$
(6.45)
also defines a real Lie algebra. An important property of this real form is that the Killing form is negative definite on it. Its signature is
$$(0\vert_{+},\dim \mathfrak {c}_{0}\vert_{-}){.}$$
(6.46)
This is an immediate consequence of the previous discussion and of the way \({{\mathfrak c}_0}\) is constructed. Hence, this real Lie algebra is compact25. For this reason, \({{\mathfrak c}_0}\) is called the “compact real form” of \({\mathfrak g}\). It is also “unique”.

6.4 Classical decompositions

6.4.1 Real forms and conjugations

The compact and split real Lie algebras constitute the two ends of a string of real forms that can be inferred from a given complex Lie algebra. As announced, this section is devoted to the systematic classification of these various real forms.

If \({{\mathfrak g}_0}\) is a real form of \({\mathfrak g}\), it defines a conjugation on \({\mathfrak g}\). Indeed we may express any \(Z \in {\mathfrak g}\) as Z = X0 + iY0 with \({X_0} \in {{\mathfrak g}_0}\) and \(i{Y_0} \in i{{\mathfrak g}_0}\), and the conjugation of \({\mathfrak g}\) with respect to \({{\mathfrak g}_0}\) is given by
$${Z}\mapsto \overline{Z}={X}_{0}-i\,{Y}_{0}{.}$$
(6.47)
Using Equation (6.3), it is immediate to verify that this involutive map is antilinear: \(\overline {\lambda Z} = \bar \lambda \bar Z\), where \(\bar \lambda\) is the complex conjugate of the complex number λ.

Conversely, if σ is a conjugation on \({\mathfrak g}\), the set \({{\mathfrak g}_\sigma}\) of elements of \({\mathfrak g}\) fixed by a provides a real form of \({\mathfrak g}\). Then σ constitutes the conjugation of \({\mathfrak g}\) with respect to \({{\mathfrak g}_\sigma}\). Thus, on \({\mathfrak g}\), real forms and conjugations are in one-to-one correspondence. The strategy used to classify and describe the real forms of a given complex simple algebra consists of obtaining all the nonequivalent possible conjugations it admits.

6.4.2 The compact real form aligned with a given real form

Let \({{\mathfrak g}_0}\) be a real form of the complex semi-simple Lie algebra \({{\mathfrak g}^{\mathbb C}} = {{\mathfrak g}_0}{\otimes _{\mathbb R}}{\mathbb C}\). Consider a compact real form \({{\mathfrak c}_0}\) of \({{\mathfrak g}^{\mathbb C}}\) and the respective conjugations τ and σ associated with \({{\mathfrak c}_0}\) and \({{\mathfrak g}_0}\). It may or it may not be that τ and σ commute. When they do, τ leaves \({{\mathfrak g}_0}\) invariant,
$$\tau(\mathfrak{g}_{0}) \subset \mathfrak{g}_{0}$$
and, similarly, σ leaves \({{\mathfrak c}_0}\) invariant,
$$\sigma (\mathfrak {c}_{0}) \subset \mathfrak {c}_{0}.$$
In that case, one says that the real form \({{\mathfrak g}_0}\) and the compact real form \({{\mathfrak c}_0}\) are “aligned”.

Alignment is not automatic. For instance, one can always de-align a compact real form by applying an automorphism to it while keeping \({{\mathfrak g}_0}\) unchanged. However, there is a theorem that states that given a real form \({{\mathfrak g}_0}\) of the complex Lie algebra \({{\mathfrak g}^{\mathbb C}}\), there is always a compact real form \({{\mathfrak c}_0}\) associated with it [93, 129]. As this result is central to the classification of real forms, we provide a proof in Appendix B, where we also prove the uniqueness of the Cartan involution.

We shall from now on always consider the compact real form aligned with the real form under study.

6.4.3 Cartan involution and Cartan decomposition

A Cartan involution θ of a real Lie algebra \({{\mathfrak g}_0}\) is an involutive automorphism such that the symmetric, bilinear form B θ defined by
$$B^\theta(X,Y) = - B(X, \theta Y)$$
(6.48)
is positive definite. If the algebra \({B^\theta}\) is compact, a Cartan involution is trivially given by the identity.
A Cartan involution θ of the real semi-simple Lie algebra \({{\mathfrak g}_0}\) yields the direct sum decomposition (called Cartan decomposition)
$$\mathfrak{g}_{0}=\mathfrak{t}_{0}\oplus\mathfrak{p}_{0},$$
(6.49)
where \({{\mathfrak k}_0}\) and \({{\mathfrak p}_0}\) are the θ-eigenspaces of eigenvalues +1 and −1. Explicitly, the decomposition of a Lie algebra element is given by
$$X=\frac{1}{2} (X + \theta[ X])+\frac{1}{2} (X - \theta[ X]){.}$$
(6.50)
The eigenspaces obey the commutation relations
$$[{\mathfrak{t}_{0}}, {\mathfrak{t}_{0}}]\subset\mathfrak{t}_{0}, \qquad [{\mathfrak{t}_{0}},{\mathfrak{p}_{0}}]\subset\mathfrak{p}_{0}, \qquad [{\mathfrak{p}_{0}}, {\mathfrak{p}_{0}}] \subset\mathfrak{t}_{0}{,}$$
(6.51)
from which we deduce that \(B({{\mathfrak k}_0},{{\mathfrak k}_0}) = 0\) because the mappings \({\rm{ab}}[{{\mathfrak k}_0}]\,{\rm{ab}}[{{\mathfrak p}_0}]\) map \({{\mathfrak p}_0}\) on \({{\mathfrak k}_0}\) and \({{\mathfrak k}_0}\) on \({{\mathfrak p}_0}\). Moreover \(\theta [{{\mathfrak k}_0}] = + {{\mathfrak k}_0}\) and \(\theta [{{\mathfrak p}_0}] = - {{\mathfrak p}_0}\), and hence \({B^\theta}({{\mathfrak k}_0},{{\mathfrak p}_0}) = 0\). In addition, since B θ is positive definite, the Killing form B is negative definite on \({{\mathfrak k}_0}\) (which is thus a compact subalgebra) but is positive definite on \({{\mathfrak p}_0}\) (which is not a subalgebra).
Define in \({{\mathfrak g}^{\mathbb C}}\) the algebra \({{\mathfrak c}_0}\) by
$$\mathfrak {c}_{0}=\mathfrak{t}_{0}\oplus i\mathfrak{p}_{0}{.}$$
(6.52)
It is clear that \({{\mathfrak c}_0}\) is also a real form of \({{\mathfrak g}^{\mathbb C}}\) and is furthermore compact since the Killing form restricted to it is negative definite. The conjugation τ that fixes \({{\mathfrak c}_0}\) is such that \(\tau (X) = X(X \in {{\mathfrak k}_0}),\,\tau (iY) = iY(Y \in {{\mathfrak p}_0})\) and hence \(\tau (Y) = - Y\,(Y \in {{\mathfrak p}_0})\). It leaves \({{\mathfrak g}_0}\) invariant, which shows that \({{\mathfrak c}_0}\) is aligned with \({{\mathfrak g}_0}\). One has
$$\mathfrak {c}_{0}=\mathfrak{t}_{0}\oplus i\mathfrak{p}_{0}, \qquad \mathfrak{t}_{0}=\mathfrak{g}_{0}\cap\mathfrak {c}_{0}, \qquad \mathfrak{p}_{0}=\mathfrak{g}_{0}\cap i\,\mathfrak {c}_{0}{.}$$
(6.53)

Conversely, let \({{\mathfrak c}_0}\) be a compact real form aligned with \({{\mathfrak g}_0}\) and τ the corresponding conjugation. The restriction θ of τ to \({{\mathfrak g}_0}\) is a Cartan involution. Indeed, one can decompose \({{\mathfrak g}_0}\) as in Equation (6.49), with Equation (6.51) holding since θ is an involution of \({{\mathfrak g}_0}\). Furthermore, one has also Equation (6.53), which shows that \({{\mathfrak k}_0}\) is compact and that B θ is positive definite.

This shows, in view of the result invoked above that an aligned compact real form always exists, that any real form possesses a Cartan involution and a Cartan decomposition. If there are two Cartan involutions, θ and θ′, defined on a real semi-simple Lie algebra, one can show that they are conjugated by an internal automorphism [93, 129]. It follows that any real semi-simple Lie algebra possesses a “unique” Cartan involution.

On the matrix algebra \({\rm{ad}}[{{\mathfrak g}_0}]\), the Cartan involution is nothing else than minus the transposition with respect to the scalar product B θ ,
$${\rm{ad}}\; \theta X=-({\rm{ad}}\; X)^{T}{.}$$
(6.54)
Indeed, remembering that the transpose of a linear operator with respect to B θ is defined by B θ (X, AY) = B θ (A T X, Y), one gets
$$\begin{array}{*{20}c} {{B^\theta}({\rm{ad}} \theta X(Y),\,Z) = - B([\theta X,Y],\,\theta Z) = B(Y,\,[\theta X,\theta Z])\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;}\\ {= B(Y,\,\theta [X,Z]) = - {B^\theta}(Y,\,{\rm{ad}} X(Z)) = - {B^\theta}({{({\rm{ad}} X)}^T}(Y),\,Z){.}} \end{array}$$
(6.55)
Since B θ is positive definite, this implies, in particular, that the operator ad Y, with \(Y \in {{\mathfrak k}_0}\), is diagonalizable over the real numbers since it is symmetric, ad Y = (ad Y) T .

An important consequence of this [93, 129] is that any real semi-simple Lie algebra can be realized as a real matrix Lie algebra, closed under transposition. One can also show [93, 129] that the Cartan decomposition of the Lie algebra of a semi-simple group can be lifted to the group via a diffeomorphism between \({{\mathfrak k}_0} \times {{\mathfrak p}_0} \mapsto {\mathcal G} = {\mathcal K}\exp [{{\mathfrak p}_0}]\), where \({\mathcal K}\) is a closed subgroup with \({{\mathfrak k}_0}\) as Lie algebra. It is this subgroup that contains all the topology of \({\mathcal G}\).

6.4.4 Restricted roots

Let \({{\mathfrak g}_0}\) be a real semi-simple Lie algebra. It admits a Cartan involution θ that allows to split it into eigenspaces \({{\mathfrak k}_0}\) of eigenvalue +1 and \({{\mathfrak p}_0}\) of eigenvalue −1. We may choose in \({{\mathfrak p}_0}\) a maximal Abelian subalgebra \({{\mathfrak a}_0}\) (because the dimension of \({{\mathfrak p}_0}\) is finite). The set \(\{{\rm{ad}}\,H\vert H \in {{\mathfrak a}_0}\}\) is a set of symmetric transformations that can be simultaneously diagonalized on ℝ. Accordingly we may decompose \({{\mathfrak g}_0}\) into a direct sum of eigenspaces labelled by elements of the dual space \(a_0^{\ast}\):
$$\mathfrak{g}_{0}=\underset{\lambda}{\bigoplus}\,{\rm{g}}_{\lambda}, \qquad {\rm{g}}_{\lambda}=\{X\in\mathfrak{g}_{0}\vert\forall H\in \mathfrak{a}_{0}:{\rm{ad}}\; H(X)=\lambda(H)\, X\}{.}$$
(6.56)

One, obviously non-vanishing, subspace is g0, which contains \({{\mathfrak a}_0}\). The other nontrivial subspaces define the restricted root spaces of \({{\mathfrak g}_0}\) with respect to \({{\mathfrak a}_0}\), of the pair \(({{\mathfrak g}_0}, \, {{\mathfrak a}_0})\). The λ that label these subspaces g λ are the restricted roots and their elements are called restricted root vectors. The set of all λ is called the restricted root system. By construction the different g λ are mutually B θ -orthogonal. The Jacobi identity implies that [g λ , g μ ] ⊂ gλ+μ, while \({{\mathfrak a}_0} \subset {{\mathfrak p}_0}\) implies that θg λ = gλ. Thus if λ is a restricted root, so is −λ.

Let \({\mathfrak m}\) be the centralizer of \({{\mathfrak a}_0}\) in \({{\mathfrak k}_0}\). The space g0 is given by
$${\rm{g}}_{0} =\mathfrak{a}_{0}\oplus\mathfrak{m}{.}$$
(6.57)
If \({{\mathfrak t}_0}\) is a maximal Abelian subalgebra of \({\mathfrak m}\), the subalgebra \({{\mathfrak h}_0} = {{\mathfrak a}_0} \oplus {{\mathfrak t}_0}\) is a Cartan subalgebra of the real algebra \({{\mathfrak g}_0}\) in the sense that its complexification \({{\mathfrak h}^{\mathbb C}}\) is a Cartan subalgebra of \({{\mathfrak g}^{\mathbb C}}\). Accordingly we may consider the set of nonzero roots Δ of \({{\mathfrak g}^{\mathbb C}}\) with respect to \({{\mathfrak h}^{\mathbb C}}\) and write
$$\mathfrak{g}^{\mathbb{C}}=\mathfrak{h}^{\mathbb{C}}\underset{\alpha \in \Delta}{\bigoplus}({\rm{g}}_{\alpha})^{\mathbb{C}}{.}$$
(6.58)
The restricted root space g λ is given by
$${\rm{g}}_{\lambda}=\mathfrak{g}_{0}\cap \underset{{\begin{array}{*{20}c} {\alpha \in \Delta}\\ {\alpha \vert {\mathfrak{a}_0} = \lambda} \end{array}}}{\bigoplus} ({\rm{g}}_{\alpha})^{\mathbb{C}}$$
(6.59)
and similarly
$$\mathfrak{m}^{\mathbb{C}}=\mathfrak{t}^{\mathbb{C}} \underset{{\begin{array}{*{20}c} {\alpha \in \Delta}\\ {\alpha \vert {\mathfrak{a}_0} = 0} \end{array}}}{\bigoplus}({\rm{g}}_{\alpha})^{\mathbb{C}}{.}$$
(6.60)
Note that the multiplicities of the restricted roots λ might be nontrivial even though the roots α are nondegenerate, because distinct roots α might yield the same restricted root when restricted to \({{\mathfrak a}_0}\).
Let us denote by Σ the subset of nonzero restricted roots and by VΣ the subspace of \(a_0^{\ast}\) that they span. One can show [93, 129] that Σ is a root system as defined in Section 4. This root system need not be reduced. As for all root systems, one can choose a way to split the roots into positive and negative ones. Let Σ+ be the set of positive roots and
$$\mathfrak{n}=\underset{\lambda\in\Sigma^{+}}{\bigoplus}\;{\rm{g}}_{\lambda}{.}$$
(6.61)
As Σ+ is finite, \({\mathfrak n}\) is a nilpotent subalgebra of \({{\mathfrak g}_0}\) and \({{\mathfrak a}_0} \oplus {\mathfrak n}\) is a solvable subalgebra.

6.4.5 Iwasawa and \({\mathcal {KAK}}\) decompositions

The Iwasawa decomposition provides a global factorization of any semi-simple Lie group in terms of closed subgroups. It can be viewed as the generalization of the Gram-Schmidt orthogonalization process.

At the level of the Lie algebra, the Iwasawa decomposition theorem states that
$$\mathfrak{g}_{0}=\mathfrak{t}_{0}\oplus\mathfrak{a}_{0}\oplus\mathfrak{n}{.}$$
(6.62)
Indeed any element X of \({{\mathfrak g}_0}\) can be decomposed as
$$X= X_{0}+\sum\limits_{\lambda} X_{\lambda}= X_{0}+\sum\limits_{\lambda\in\Sigma^{+}}(X_{-\lambda}+\theta X_{-\lambda})+ \sum\limits_{\lambda\in\Sigma^{+}}(X_{\lambda}-\theta X_{-\lambda}){.}$$
(6.63)
The first term X0 belongs to \(g = {{\mathfrak a}_0} \oplus {\mathfrak m} \subset {{\mathfrak {a}_0}} \oplus {{\mathfrak {k}_0}}\), while the second term belongs to \({{\mathfrak k}_0}\), the eigenspace subspace of θ-eigenvalue +1. The third term belongs to \({\mathfrak n}\) since θX−λgλ. The sum is furthermore direct. This is because one has obviously \({{\mathfrak {k}_0}} \cap {{\mathfrak {a}_0}} = 0\) as well as \({{\mathfrak {a}_0}} \cap {\mathfrak n} = 0\). Moreover, \({{\mathfrak {k}_0}} \cap {\mathfrak n}\) also vanishes because \(\theta {\mathfrak n} \cap {\mathfrak n} = 0\) as a consequence of \(\theta {\mathfrak n} = {\oplus _{\lambda \in \Sigma}} + {g_{- \lambda}}\).
The Iwasawa decomposition of the Lie algebra differs from the Cartan decomposition and is tilted with respect to it, in the sense that \({\mathfrak n}\) is neither in \({{\mathfrak k}_0}\) nor in \({{\mathfrak p}_0}\). One of its virtues is that it can be elevated from the Lie algebra \({{\mathfrak g}_0}\) to the semi-simple Lie group \({\mathcal G}\). Indeed, it can be shown [93, 129] that the map
$$(k,a,n) \in {\mathcal K} \times {\mathcal A} \times {\mathcal N} \mapsto k\,a\,n \in {\mathcal G}$$
(6.64)
is a global diffeomorphism. Here, the subgroups \({\mathcal K},\,{\mathcal A}\) and \({\mathcal N}\) have respective Lie algebras \({\mathfrak k_0},\,{\mathfrak a_0}\). This decomposition is “unique”.

There is another useful decomposition of \({\mathcal G}\) in terms of a product of subgroups. Any two generators of \({{\mathfrak p}_0}\) are conjugate via internal automorphisms of the compact subgroup \({\mathcal K}\). As a consequence writing an element \(g \in {\mathcal G}\) as a product \(g = k\,{\rm{Exp}}[{{\mathfrak p}_0}]\), we may write \({\mathcal G} = {\mathcal K}{\mathcal A}{\mathcal K}\), which constitutes the so-called \({\mathcal K}{\mathcal A}{\mathcal K}\) decomposition of the group (also sometimes called the Cartan decomposition of the group although it is not the exponention of the Cartan decomposition of the algebra). Here, however, the writing of an element of \({\mathcal G}\) as product of elements of \({\mathcal K}\) and \({\mathcal A}\) is, in general, not unique.

6.4.6 θ-stable Cartan subalgebras

As in the previous sections, \({{\mathfrak g}_0}\) is a real form of the complex semi-simple algebra \({\mathfrak g}\), σ denotes the conjugation it defines, τ the conjugation that commutes with σ, \({{\mathfrak c}_0}\) the associated compact aligned real form of \({\mathfrak g}\) and θ the Cartan involution. It is also useful to introduce the involution of \({\mathfrak g}\) given by the product στ of the commuting conjugations. We denote it also by θ since it reduces to the Cartan involution when restricted to \({{\mathfrak g}_0}\). Contrary to the conjugations σ and τ, θ is linear over the complex numbers. Accordingly, if we complexify the Cartan decomposition \({{\mathfrak g}_0} = {{\mathfrak k}_0} \oplus {{\mathfrak p}_0}\), to
$${\mathfrak g} = {\mathfrak k} \oplus {\mathfrak p}$$
(6.65)
with \({\mathfrak k} = {{\mathfrak k}_0}{\oplus _{\mathbb R}}{\mathbb C} = {{\mathfrak k}_0} \oplus i{{\mathfrak k}_0}\) and \({\mathfrak p} = {{\mathfrak p}_0}{\oplus _{\mathbb R}}{\mathbb C} = {{\mathfrak p}_0} \oplus i{{\mathfrak p}_0}\), the involution θ fixes \({\mathfrak k}\) pointwise while θ(X)=−X \(X \in {\mathfrak p}\).
Let \({{\mathfrak h}_0}\) be a θ-stable Cartan subalgebra of \({{\mathfrak g}_0}\), i.e., a subalgebra such that (i) \(\theta ({{\mathfrak h}_0}) \subset {{\mathfrak h}_0}\), and (ii) \({\mathfrak h} \equiv {\mathfrak h}_0^{\mathbb C}\) is a Cartan subalgebra of the complex algebra \({\mathfrak g}\). One can decompose \({{\mathfrak h}_0}\) into compact and noncompact parts,
$${{\mathfrak h}_0} = {{\mathfrak t}_0} \oplus {{\mathfrak a}_0},\quad {{\mathfrak t}_0} = {{\mathfrak h}_0} \cap {{\mathfrak k}_0},\quad {{\mathfrak a}_0} = {{\mathfrak h}_0} \cap {{\mathfrak p}_0}.$$
(6.66)

We have seen that for real Lie algebras, the Cartan subalgebras are not all conjugate to each other; in particular, even though the dimensions of the Cartan subalgebras are all equal to the rank of \({\mathfrak g}\), the dimensions of the compact and noncompact subalgebras depend on the choice of \({{\mathfrak h}_0}\). For example, for \({\mathfrak {sl}(2,\,\mathbb R)}\), one may take \({{\mathfrak h}_0} = {\mathbb R}t\), in which case \({{\mathfrak t}_0} = 0,\,{{\mathfrak a}_0} = {{\mathfrak h}_0}\). Or one may take \({{\mathfrak h}_0} = {\mathbb R}{\tau ^y}\), in which case \({{\mathfrak t}_0} = {{\mathfrak h}_0},\,{{\mathfrak a}_0} = 0\).

One says that the θ-stable Cartan subalgebra \({{\mathfrak h}_0}\) is maximally compact if the dimension of its compact part \({{\mathfrak t}_0}\) is as large as possible; and that it is maximally noncompact if the dimension of its noncompact part \({{\mathfrak a}_0}\) is as large as possible. The θ-stable Cartan subalgebra \({{\mathfrak h}_0} = {{\mathfrak t}_0} \oplus {{\mathfrak a}_0}\) used above to introduce restricted roots, where \({{\mathfrak a}_0}\) is a maximal Abelian subspace of \({{\mathfrak p}_0}\) and \({{\mathfrak t}_0}\) a maximal Abelian subspace of its centralizer \({\mathfrak m}\), is maximally noncompact. If \({\mathfrak m} = 0\), the Lie algebra \({{\mathfrak g}_0}\) constitutes a split real form of \({{\mathfrak g}^{\mathcal C}}\). The real rank of \({{\mathfrak g}_0}\) is the dimension of its maximally noncompact Cartan subalgebras (which can be shown to be conjugate, as are the maximally compact ones [129]).

6.4.7 Real roots — Compact and non-compact imaginary roots

Consider a general θ-stable Cartan subalgebra \({{\mathfrak h}_0} = {{\mathfrak t}_0} \oplus {{\mathfrak a}_0}\), which need not be maximally compact or maximally non compact. A consequence of Equation (6.54) is that the matrices of the real linear transformations ad H are real symmetric for \(H \in {{\mathfrak a}_0}\) and real antisymmetric for \(H \in {{\mathfrak t}_0}\). Accordingly, the eigenvalues of ad H are real (and ad H can be diagonalized over the real numbers) when \(H \in {{\mathfrak a}_0}\), while the eigenvalues of ad H are imaginary (and ad H cannot be diagonalized over the real numbers although it can be diagonalized over the complex numbers) when \(H \in {{\mathfrak t}_0}\).

Let α be a root of \({\mathfrak g}\), i.e., a non-zero eigenvalue of ad \({\mathfrak h}\) where \({\mathfrak h}\) is the complexification of the θ-stable Cartan subalgebra \({{\mathfrak h}_0}\). As the values of the roots acting on a given H are the eigenvalues of ad H, we find that the roots are real on \({{\mathfrak a}_0}\) and imaginary on \({{\mathfrak t}_0}\). One says that a root is real if it takes real values on \({{\mathfrak h}_0} = {{\mathfrak t}_0} \oplus {{\mathfrak a}_0}\), i.e., if it vanishes on \({{\mathfrak t}_0}\). It is imaginary if it takes imaginary values on \({{\mathfrak h}_0}\), i.e., if it vanishes on \({{\mathfrak n}_0}\), and complex otherwise. These notions of “real” and “imaginary” roots should not be confused with the concepts with similar terminology introduced in Section 4 in the context of non-finite-dimensional Kac-Moody algebras.

If \({{\mathfrak h}_0}\) is a θ-stable Cartan subalgebra, its complexification \({\mathfrak h} = {{\mathfrak h}_0} {\oplus_{\mathbb R}}{\mathbb C} {{\mathfrak h}_0} \oplus i{{\mathfrak h}_0}\) is stable under the involutive authormorphism θ = τσ. One can extend the action of θ from \({\mathfrak h}\) to \({{\mathfrak h}^{\ast}}\) by duality. Denoting this transformation by the same symbol θ, one has
$$\forall H \in {\mathfrak h}\;{\rm{and}}\;\forall \alpha \in {{\mathfrak h}^\ast},\qquad \theta (\alpha)(H) = \alpha ({\theta ^{- 1}}(H)),$$
(6.67)
or, since θ2 = 1,
$$\theta (\alpha)(H) = \alpha (\theta H){.}$$
(6.68)
Let E α be a nonzero root vector associated with the root α and consider the vector θE α . One has
$$[H,\theta {E_\alpha}] = \theta \;[\theta H,{E_\alpha}] = \alpha (\theta H)\,\theta {E_\alpha} = \theta (\alpha)(H)\,\theta {E_\alpha},$$
(6.69)
i.e., θ(g α ) = gθ(α) because the roots are nondegenerate, i.e., all root spaces are one-dimensional.
Consider now an imaginary root α. Then for all \(h \in {{\mathfrak h}_0}\) and \(a \in {{\mathfrak a}_0}\) we have α(h + a) = α(h), while θ(α) (h + a) = α(θ(h + a)) = α(ha) = α(h); accordingly α = θ(α). Moreover, as the roots are nondegenerate, one has θE α = ±E α . Writing E α as
$${E_\alpha} = {X_\alpha} + i\,{Y_\alpha}\qquad {\rm{with}}{X_\alpha},\,{Y_\alpha} \in {\mathfrak g}_0,$$
(6.70)
it is easy to check that θE α = +E α implies that X α and Y α belong to \({{\mathfrak k}_0}\), while both are in \({{\mathfrak p}_0}\) if θE α = −E α . Accordingly, g α is completely contained either in \({\mathfrak k} = {{\mathfrak k}_0} \oplus i{{\mathfrak k}_0}\) or in \({\mathfrak p} = {{\mathfrak p}_0} \oplus i{{\mathfrak p}_0}\). If \({g_\alpha} \subset {\mathfrak k}\), the imaginary root is said to be compact, and if \({g_\alpha} \subset {\mathfrak p}\) it is said to be noncompact.

6.4.8 Jumps between Cartan subalgebras — Cayley transformations

Suppose that β is an imaginary noncompact root. Consider a β-root vector \({E_\beta} \in {g_\beta} \subset {\mathfrak p}\). If this root is expressed according to Equation (6.70), then its conjugate, with respect to (the conjugation σ defined by) \({{\mathfrak g}_0}\), is
$$\sigma {E_\beta} = {X_\beta} - i\,{Y_\beta}\qquad {\rm{with}}\;{X_\beta},\,{Y_\beta} \in {{\mathfrak p}_0}.$$
(6.71)
It belongs to gβ because (using \(\forall H \in {{\mathfrak h}_0}:\sigma H = H\))
$$[H,\sigma {E_\beta}] = \sigma [\sigma H,{E_\beta}] = \overline {\beta (\sigma H)} \,\sigma {E_\beta} = - \beta (H)\,\sigma {E_\beta}.$$
(6.72)
Hereafter, we shall denote σE β by \({{\bar E}_\beta}\). The commutator
$$[{E_\beta},{\bar E_\beta}] = B({E_\beta},\,{\bar E_\beta})\,{H_\beta}$$
(6.73)
belongs to i \(i{\mathfrak k}_0\) since \(\sigma ([{E_\beta},\,{{\bar E}_\beta}]) = [{E_\beta},\,{{\bar E}_\beta}] = - [{E_\beta},\,{{\bar E}_\beta}]\) and can be written, after a renormalization of the generators E β , as
$$[{E_\beta},{\bar E_\beta}] = {2 \over {(\beta \vert \beta)}}\,{H_\beta} = H_\beta\prime \qquad \in i\,{{\mathfrak k}_0}.$$
(6.74)
Indeed as \({E_\beta} \in {\mathfrak p}\), we have \({{\bar E}_\beta} \in {\mathfrak p}\) and thus \(\theta {{\bar E}_\beta} = - {{\bar E}_\beta}\). This implies
$$B({E_\beta}\,,{\bar E_\beta}) = - B({E_\beta},\,\theta {\bar E_\beta}) = {B^\theta}({E_\beta},{\bar E_\beta}) > 0.$$
The three generators \(\{{H_\beta},\,{E_\beta},\,{{\bar E}_\beta}\}\) therefore define an \({\mathfrak {sl}(2,\,\mathbb C)}\) subalgebra:
$$[{E_\beta},{\bar E_\beta}] = {H_{{\beta {\prime}}}},\qquad [{H_{{\beta {\prime}}}},{E_\beta}] = 2\,{E_\beta},\qquad [{H_{{\beta {\prime}}}},{\bar E_\beta}] = - 2\,{\bar E_\beta}.$$
(6.75)
We may change the basis and take
$$h = {E_\beta} + {\bar E_\beta},\qquad e = {i \over 2}({E_\beta} - {\bar E_\beta} - {H_{{\beta{\prime}}}}),\qquad f = {i \over 2}({E_\beta} - {\bar E_\beta} + {H_{{\beta\prime}}}),$$
(6.76)
whose elements belong to \({{\mathfrak g}_0}\) (since they are fixed by σ) and satisfy the commutation relations (6.8)
$$[e,f] = h,\qquad [h,e] = 2\,e,\qquad [h,f] = - 2\,f.$$
(6.77)
The subspace
$${{\mathfrak h}\prime_0} = \ker (\beta \vert {{\mathfrak h}_0}) \oplus {\mathbb R}h$$
(6.78)
constitutes a new real Cartan subalgebra whose intersection with \({{\mathfrak p}_0}\) has one more dimension.
Conversely, if β is a real root then θ(β) = −β. Let E β be a root vector. Then \({{\bar E}_\beta}\) is also in g β and hence proportional to E β . By adjusting the phase of E β , we may assume that E β belongs to \({{\mathfrak g}_0}\). At the same time, θE β , also in \({{\mathfrak g}_0}\), is an element of gβ. Evidently, B(E β , θE β ) = −B θ (E β , E β ) is negative. Introducing Hβ = 2/(β∣β)H β (which is in \({{\mathfrak p}_0}\)), we obtain the \({\mathfrak {sl}(2,\,\mathbb R)}\) commutation relations
$$[{E_\beta},\theta {E_\beta}] = - {H_{\beta \prime}}\, \in {{\mathfrak p}_0},\qquad [{H_{\beta \prime}},{E_\beta}] = 2\,{E_\beta},\qquad [{H_{\beta \prime}},\theta {E_\beta}] = - 2\,\theta {E_\beta}.$$
(6.79)
Defining the compact generator E β + θE β , which obviously belongs to \({{\mathfrak g}_0}\), we may build a new Cartan subalgebra of \({{\mathfrak g}_0}\):
$${{\mathfrak h}\prime_0} = \ker (\beta \vert {{\mathfrak h}_0}) \oplus {\mathbb R}({E_\beta} + \theta {E_\beta}),$$
(6.80)
whose noncompact subspace is now one dimension less than previously.

These two kinds of transformations — called Cayley transformations — allow, starting from a θ-stable Cartan subalgebra, to transform it into new ones with an increasing number of noncompact dimensions, as long as noncompact imaginary roots remain; or with an increasing number of compact dimensions, as long as real roots remain. Exploring the algebra in this way, we obtain all the Cartan subalgebras up to conjugacy. One can prove that the endpoints are maximally noncompact and maximally compact, respectively.

Theorem: Let \({{\mathfrak h}_0}\) be a θ stable Cartan subalgebra of \({{\mathfrak g}_0}\). Then there are no noncompact imaginary roots if and only if \({{\mathfrak h}_0}\) is maximally noncompact, and there are no real roots if and only if \({{\mathfrak h}_0}\) is maximally compact [129].

For a proof of this, note that we have already proven that if there are imaginary noncompact (respectively, real) roots, then \({{\mathfrak h}_0}\) is not maximally noncompact (respectively, compact). The converse is demonstrated in [129].

6.5 Vogan diagrams

Let \({{\mathfrak g}_0}\) be a real semi-simple Lie algebra, \({\mathfrak g}\) its complexification, θ a Cartan involution leading to the Cartan decomposition
$${{\mathfrak g}_0} = {{\mathfrak k}_0} \oplus {{\mathfrak p}_0},$$
(6.81)
and \({{\mathfrak h}_0}\) a Cartan θ-stable subalgebra of \({{\mathfrak g}_0}\). Using, if necessary, successive Cayley transformations, we may build a maximally compact θ-stable Cartan subalgebra \({{\mathfrak h}_0} = {{\mathfrak t}_0} \oplus {{\mathfrak a}_0}\), with complexification \({\mathfrak h} = {\mathfrak t} \oplus {\mathfrak a}\). As usual we denote by Δ the set of (nonzero) roots of \({\mathfrak g}\) with respect to \({\mathfrak h}\). This set does not contain any real root, the compact dimension being assumed to be maximal.

From Δ we may define a positive subset Δ+ by choosing the first set of indices from a basis of \(i{{\mathfrak t}_0}\), and then the next set from a basis of \({{\mathfrak a}_0}\). Since there are no real roots, the roots in Δ+ have at least one non-vanishing component along \(i{{\mathfrak t}_0}\), and the first non-zero one of these components is strictly positive. Since θ = +1 on \({{\mathfrak t}_0}\), and since there are no real roots: θΔ+ = Δ+. Thus θ permutes the simple roots, fixes the imaginary roots and exchanges in 2-tuples the complex roots: it constitutes an involutive automorphism of the Dynkin diagram of \({\mathfrak g}\).

A Vogan diagram is associated to the triple \(({{\mathfrak g}_0},\,{{\mathfrak h}_{0,\,}}{\Delta ^ +})\) as follows. It corresponds to the standard Dynkin diagram of Δ+, with additional information: the 2-element orbits under θ are exhibited by joining the correponding simple roots by a double arrow and the 1-element orbit is painted in black (respectively, not painted), if the corresponding imaginary simple root is noncompact (respectively, compact).

6.5.1 Illustration — The \({\mathfrak {sl}}(5,\,{\mathbb C})\) case

The complex Lie algebra \({\mathfrak {sl}}(5,\,{\mathbb C})\) can be represented as the algebra of traceless 5 × 5 complex matrices, the Lie bracket being the usual commutator. It has dimension 24. In principle, in order to compute the Killing form, one needs to handle the 24 × 24 matrices of the adjoint representation. Fortunately, the uniqueness (up to an overall factor) of the bi-invariant quadratic form on a simple Lie algebra leads to the useful relation
$$B(X,Y) = {\rm{Tr}}({\rm{ad}}\;X\;{\rm{ad}}\;Y) = 10\;{\rm{Tr}}(XY).$$
(6.82)
The coefficient 10 appearing in this relation is known as the Coxeter index of \({\mathfrak {sl}}(5,\,{\mathbb C})\).
A Cartan-Weyl basis is obtained by taking the 20 nilpotent generators \({K^p}_q\) (with pq) corresponding to matrices, all elements of which are zero except the one located at the intersection of row p and column q, which is equal to 1,
$$({K^p}_q)_\beta ^\alpha = {\delta ^{\alpha \,p}}{\delta _{\beta \,q}}$$
(6.83)
and the four diagonal ones,
$$\begin{array}{*{20}c} {{H_1} = \left({\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array}} \right),} & {{H_2} = \left({\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & {- 1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array}} \right)} \\ {{H_3} = \left({\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array}} \right),} & {{H_4} = \left({\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & {- 1} \\ \end{array}} \right)} \\\end{array}$$
(6.84)
which constitute a Cartan subalgebra \({\mathfrak h}\).
The root space is easily described by introducing the five linear forms ϵ p , acting on diagonal matrices d = diag(d1, …, d5) as follows:
$${\epsilon_p}(d) = {d_p}.$$
(6.85)
In terms of these, the dual space \({{\mathfrak h}^{\ast}}\) of the Cartan subalgebra may be identified with the subspace
$$\left\{{\epsilon= \sum\limits_p {{A^p}} \,\epsilon_{p}\vert \sum\limits_p {{A^p}} = 0} \right\}.$$
(6.86)
The 20 matrices \({K^p}_q\) are root vectors,
$$[{H_k},{K^p}_q] = ({\epsilon_p}[{H_k}] - {\epsilon_q}[{H_k}]){K^p}_q,$$
(6.87)
i.e., \({K^p}_q\) is a root vector associated to the root ϵ p ϵ q .
6.5.1.1 \({\mathfrak {sl}}(5,\,{\mathbb R})\) and \({\mathfrak {su}}(5)\)
By restricting ourselves to real combinations of these generators we obtain the real Lie algebra \({\mathfrak {sl}}(5,\,{\mathbb R})\). The conjugation η that it defines on \({\mathfrak {sl}}(5,\,{\mathbb C})\) is just the usual complex conjugation. This \({\mathfrak {sl}}(5,\,{\mathbb R})\) constitutes the split real form \({{\mathfrak s}_0}\) of \({\mathfrak {sl}}(5,\,{\mathbb C})\). Applying the construction given in Equation (6.45) to the generators of \({\mathfrak {sl}}(5,\,{\mathbb R})\), we obtain the set of antihermitian matrices
$$i\,{H_k},\qquad {K^p}_q - {K^q}_p,\qquad i({K^p}_q + {K^q}_p)\qquad (p > q),$$
(6.88)
defining a basis of the real subalgebra \({\mathfrak {su}}(5)\). This is the compact real form \(i{{\mathfrak c}_0}\) of \({\mathfrak {sl}}(5,\,{\mathbb C})\). The conjugation associated to this algebra (denoted by τ) is minus the Hermitian conjugation,
$$\tau (X) = - {X^\dagger}.$$
(6.89)
Since [η, τ] =0, τ induces a Cartan involution θ on \({\mathfrak {sl}}(5,\,{\mathbb R})\), providing a Euclidean form on the previous \({\mathfrak {sl}}(5,\,{\mathbb R})\) subalgebra
$${B^\theta}(X,\,Y) = 10\;\,{\rm{Tr}}(X{Y^t}),$$
(6.90)
which can be extended to a Hermitian form on \({\mathfrak {sl}}(5,\,{\mathbb C})\),
$${B^\theta}(X,\,Y) = 10\;\,{\rm{Tr}}(X{Y^\dagger}),$$
(6.91)
Note that the generators i H k and \(i({k^p}_q + {k^q}_p)\) are real generators (although described by complex matrices) since, e.g., \({(i{H_k})^\dagger} = - iH_k^\dagger\), i.e., \(\tau (i{H_k}) = i{H_k}\).
6.5.1.2 The other real forms
The real forms of \({\mathfrak {sl}}(5,\,{\mathbb C})\) that are not isomorphic to \({\mathfrak {sl}}(5,\,{\mathbb R})\) or \({\mathfrak {su}}(5)\) are isomorphic either to \({\mathfrak {su}}(3,\,2)\) or \({\mathfrak {su}}(4,\,1)\). In terms of matrices these algebras can be represented as
$$\begin{array}{*{20}c} {X = \left({\begin{array}{*{20}c} A & \Gamma \\ {{\Gamma ^\dagger}} & B \\ \end{array}} \right)\quad \quad {\rm{where}}\;A = - {A^\dagger} \in {{\mathbb C}^{p \times p}},\quad B = - {B^\dagger} \in {{\mathbb C}^{q \times q}},} \\ {{\rm{Tr}}\;A + {\rm{Tr}}\;B = 0,\quad \Gamma \in {{\mathbb C}^{p \times q}}\quad \;{\rm{with}}\;p = 3\;{\rm{(respectively}}\;{\rm{4)}}\;{\rm{and}}\;q = 2\;{\rm{(respectively}}\;{\rm{1)}}.} \\\end{array}$$
(6.92)
We shall call these ways of describing \({\mathfrak {su}}(p,\,q)\) the “natural” descriptions of \({\mathfrak {su}}(p,\,q)\). Introducing the diagonal matrix
$${I_{p,\, q}} = \left({\begin{array}{*{20}c} {I{d^{p \times p}}\quad \quad \quad \quad} \\ {\quad \quad \quad - I{d^{q \times q}}} \\\end{array}} \right),$$
(6.93)
the conjugations defined by these subalgebras are given by:
$${\sigma _{p, q}}(X) = - {I_{p, q}}{X^\dagger}{I_{p, q}}.$$
(6.94)
6.5.1.3 Vogan diagrams
The Dynkin diagram of \({\mathfrak {sl}}(5,\,{\mathbb C})\) is of A4 type (see Figure 26).
Figure 26

The A4 Dynkin diagram.

Let us first consider an \({\mathfrak {su}}(3,\,2)\) subalgebra. Diagonal matrices define a Cartan subalgebra whose all elements are compact. Accordingly all associated roots are imaginary. If we define the positive roots using the natural ordering ϵ1 > ϵ2 > ϵ3 > ϵ4 > ϵ5, the simple roots α1 = ϵ1 − ϵ2, α2 = ϵ2ϵ3, α4 = ϵ4ϵ5 are compact but α3 = ϵ3ϵ4 is noncompact. The corresponding Vogan diagram is displayed in Figure 27.
Figure 27

A Vogan diagram associated to \({\mathfrak {su}}(3,\,2)\).

However, if instead of the natural order we define positive roots by the rule ϵ1 > ϵ2 > ϵ4 > ϵ5 > ϵ3, the simple positive roots are \({\tilde \alpha _1} = {\epsilon _1} - {\epsilon _2}\) and \({\tilde \alpha _3} = {\epsilon _4} - {\epsilon _5}\) which are compact, and \({\tilde \alpha _2} = {\epsilon _2} - {\epsilon _4}\) and \({\tilde \alpha _4} = {\epsilon _5} - {\epsilon _3}\) which are noncompact. The associated Vogan diagram is shown in Figure 28.
Figure 28

Another Vogan diagram associated to \({\mathfrak {su}}(3,\,2)\).

Alternatively, the choice of order ϵ1 > ϵ5 > ϵ3 > ϵ4 > ϵ2 leads to the diagram in Figure 29.
Figure 29

Yet another Vogan diagram associated to \({\mathfrak {su}}(3,\,2)\).

There remain seven other possibilities, all describing the same subalgebra \({\mathfrak {su}}(3,\,2)\). These are displayed in Figure 30.
Figure 30

The remaining Vogan diagrams associated to \({\mathfrak {su}}(3,\,2)\).

In a similar way, we obtain four different Vogan diagrams for \({\mathfrak {su}}(4,\,1)\), displayed in Figure 31.
Figure 31

The four Vogan diagrams associated to \({\mathfrak {su}}(4,\,1)\).

Finally we have two non-isomorphic Vogan diagrams associated with \({\mathfrak {su}}(5)\) and \({\mathfrak {sl}}(5,\,{\mathbb R})\). These are shown in Figure 32.
Figure 32

The Vogan diagrams for \({\mathfrak {su}}(5)\) and \({\mathfrak {sl}}(5,\,{\mathbb R})\).

6.5.2 The Borel and de Siebenthal theorem

As we just saw, the same real Lie algebra may yield different Vogan diagrams only by changing the definition of positive roots. But fortunately, a theorem of Borel and de Siebenthal tells us that we may always choose the simple roots such that at most one of them is noncompact [129]. In other words, we may always assume that a Vogan diagram possesses at most one black dot.

Furthermore, assume that the automorphism associated with the Vogan diagram is the identity (no complex roots). Let {α p } be the basis of simple roots and {Λ q } its dual basis, i.e., (Λ q α p ) = δ p q . Then the single painted simple root α p may be chosen so that there is no q with (Λ p − Λ q ∣Λ q ) > 0. This remark, which limits the possible simple root that can be painted, is particularly helpful when analyzing the real forms of the exceptional groups. For instance, from the Dynkin diagram of E8 (see Figure 33), it is easy to compute the dual basis and the matrix of scalar products Bp q = (Λp − Λq ∣Λq).
Figure 33

The Dynkin diagram of E 8 . Seen as a Vogan diagram, it corresponds to the maximally compact form of E8.

We obtain
$$({B_{p\, q}}) = \left({\begin{array}{*{20}c} {- 0} & {- 7} & {- 20} & {- 12} & {- 6} & {- 2} & {- 0} & {- 3} \\ {- 3} & {- 0} & {- 10} & {- 4} & {- 0} & {- 2} & {- 2} & {- 2} \\ {- 6} & {- 6} & {- 0} & {- 4} & {- 6} & {- 6} & {- 4} & {- 7} \\ {- 4} & {- 2} & {- 6} & {- 0} & {- 3} & {- 4} & {- 3} & {- 4} \\ {- 2} & {- 2} & {- 12} & {- 5} & {- 0} & {- 2} & {- 2} & {- 1} \\ {- 0} & {- 6} & {- 18} & {- 10} & {- 4} & {- 0} & {- 1} & {- 2} \\ {- 2} & {- 10} & {- 24} & {- 15} & {- 8} & {- 3} & {- 0} & {- 5} \\ {- 1} & {- 4} & {- 15} & {- 8} & {- 3} & {- 0} & {- 1} & {- 0} \\\end{array}} \right),$$
(6.95)
from which we see that there exist, besides the compact real form, only two other non-isomorphic real forms of E8, described by the Vogan diagrams in Figure 3426.
Figure 34

Vogan diagrams of the two different noncompact real forms of E8: E8(−24) and E8(8). The lower one corresponds to the split real form.

6.5.3 Cayley transformations in su(3, 2)

Let us now illustrate the Cayley transformations. For this purpose, consider again \({\mathfrak {su}}(3,\,2)\) with the imaginary diagonal matrices as Cartan subalgebra and the natural ordering of the ϵ k defining the positive roots. As we have seen, α3 = ϵ3ϵ4 is an imaginary noncompact root. The associated \({\mathfrak {sl}}(2,\,{\mathbb C})\) generators are
$${E_{{\alpha _3}}} = K_4^3,\quad \overline {{E_{{\alpha _3}}}} = \sigma K_4^3 = K_3^4,\quad i {H_3}.$$
(6.96)
From the action of α3 on the Cartan subalgebra D = span{i H k , k = 1, …, 4}, we may check that
$$\ker ({\alpha _3}\vert D) = {\rm{span}}\{i{H_1},\, i(2{H_2} + {H_3}), \,i(2{H_4} + {H_3})\} ,$$
(6.97)
and that \({H\prime} = ({E_{{\alpha _3}}} + \overline {{E_{{\alpha _3}}}}) = (K_4^3 + K_3^4)\) is such that θH′ = −H′ and σH′ = H′. Moreover H′ commutes with ker(α3D). Thus
$$C = \ker ({\alpha _3}\vert D) \oplus {\mathbb R},\,{H\prime}$$
(6.98)
constitutes a θ-stable Cartan subalgebra with one noncompact dimension H′. Indeed, we have B(H′, H′) = 20. If we compute the roots with respect to this new Cartan subalgebra, we obtain twelve complex roots (expressed in terms of their components in the basis dual to the one implicitly defined by Equations (6.97) and (6.98),
$$\pm (i,\, - 3i,\,i, \pm 1),\quad \pm (0,\,i,\, - 3i,\, \pm 1),\quad \pm (i,\,i,\, - i,\, \pm 1),$$
(6.99)
six imaginary roots
$$\pm i(2,\, - 2,\,0,\,0),\quad \pm i(1,\, - 2,\, - 2,\,0),\quad \pm i(1,\,0,\,2,\,0),$$
(6.100)
and a pair of real roots ±(0, 0, 0, 2).
Let us first consider the Cayley transformation obtained using, for instance, the real root (0, 0, 0, 2). An associated root vector, belonging to \({{\mathfrak g}_0}\), reads
$$E = \left({\begin{array}{*{20}c} 0 & 0 & 0 & {\quad 0} & 0 \\ 0 & 0 & 0 & {\quad 0} & 0 \\ 0 & 0 & {{i \over 2}} & {- {i \over 2}} & 0 \\ 0 & 0 & {{i \over 2}} & {- {i \over 2}} & 0 \\ 0 & 0 & 0 & {\quad 0} & 0 \\\end{array}} \right).$$
(6.101)
The corresponding compact Cartan generator is
$$h = \left({\begin{array}{*{20}c} 0 & 0 & 0 & {\;\;0} & 0 \\ 0 & 0 & 0 & {\;\;0} & 0 \\ 0 & 0 & i & {\;\;0} & 0 \\ 0 & 0 & 0 & {- i} & 0 \\ 0 & 0 & 0 & {\;\;0} & 0 \\\end{array}} \right),$$
(6.102)
which, together with the three generators in Equation (6.97), provide a compact Cartan subalgebra of \({\mathfrak {su}}(3,\,2)\).
If we consider instead the imaginary roots, we find for instance that \(K_2^5 = - \tilde \theta K_2^5\) is a noncompact complex root vector corresponding to the root β = i(1, −2, −2, 0). It provides the noncompact generator \(K_5^2 = + \,K_2^5\) which, together with
$$\ker (\beta \vert C) = {\rm{span}}\{i(2 {H_1} + 2 {H_2} + {H_3}),\, i(2 {H_1} + {H_3} + 2 {H_4}),\, K_4^3 + K_3^4\} ,$$
(6.103)
generates a maximally noncompact Cartan subalgebra of \({\mathfrak {su}}(3,\,2)\). A similar construction can be done using, for instance, the roots ±i(1, 0, 2, 0), but not with the roots ±i(2, −2, 0, 0) as their corresponding root vectors \(K_2^1\) and \(K_1^2\) are fixed by \(\tilde \theta\) and thus are compact.

6.5.4 Reconstruction

We have seen that every real Lie algebra leads to a Vogan diagram. Conversely, every Vogan diagram defines a real Lie algebra. We shall sketch the reconstruction of the real Lie algebras from the Vogan diagrams here, referring the reader to [129] for more detailed information.

Given a Vogan diagram, the reconstruction of the associated real Lie algebra proceeds as follows. From the diagram, which is a Dynkin diagram with extra information, we may first construct the associated complex Lie algebra, select one of its Cartan subalgebras and build the corresponding root system. Then we may define a compact real subalgebra according to Equation (6.45).

We know the action of θ on the simple roots. This implies that the set Δ of all roots is invariant under θ. This is proven inductively on the level of the roots, starting from the simple roots (level 1). Suppose we have proven that the image under θ of all the positive roots, up to level n are in Δ. If γ is a root of level n +1, choose a simple root α such that (γα) > 0. Then the Weyl transformed root s α γ = β is also a positive root, but of smaller level. Since θ(α) and θ(β) are then known to be in Δ, and since the involution acts as an isometry, θ(γ) = sθ(α)(θ(β)) is also in Δ.

One can transfer by duality the action of θ on \({{\mathfrak h}^{\ast}}\) to the Cartan subalgebra \({\mathfrak h}\), and then define its action on the root vectors associated to the simple roots according to the rules
$$\theta {E_\alpha} = \left\{{\begin{array}{*{20}c} {\;{E_\alpha}\quad \quad \quad \quad \quad {\rm{if}}\,\alpha \,{\rm{is}}\,{\rm{unpainted}}\,{\rm{and}}\,{\rm{invariant}},\quad \quad \quad \quad \quad} \\ {- {E_\alpha}\quad \quad \;\quad \quad \;\;{\rm{if}}\,\alpha \,{\rm{is}}\,{\rm{painted}}\,{\rm{and}}\,{\rm{invariant}},\quad \quad \quad \quad \quad \quad} \\ {- {E_{\theta [\alpha ]}}\quad \quad \;\;\quad {\rm{if}}\,\alpha \,{\rm{belongs}}\,{\rm{to}}\,{\rm{a}}\,{\rm{2 - cycle}}.\quad \quad \quad \quad \quad \quad \quad} \\ \end{array}} \right.$$
(6.104)
These rules extend θ to an involution of \({\mathfrak g}\).
This involution is such that θE α = a α Eθ[α], with a α = ±1 27. Furthermore it globally fixes \({{\mathfrak c}_0},\,\theta {{\mathfrak c}_0} = {{\mathfrak c}_0}\). Let \({\mathfrak k}\) and \({\mathfrak p}\) be the +1 or −1 eigenspaces of θ in \({\mathfrak g} = {\mathfrak k} \oplus {\mathfrak p}\). Define \({{\mathfrak k}_0} = {{\mathfrak c}_0} \cap {\mathfrak k}\) and \({{\mathfrak p}_0} = i({{\mathfrak c}_0} \cap {\mathfrak p})\) so that \({{\mathfrak c}_0} = {{\mathfrak k}_0} \oplus i\,{{\mathfrak p}_0}\). Set
$${\mathfrak{g}}_{0}={\mathfrak{k}}_{0}\oplus{\mathfrak{p}}_{0}.$$
(6.105)
Using \(\theta {{\mathfrak c}_0} = {{\mathfrak c}_0}\), one then verifies that \(i{{\mathfrak g}_0}\) constitutes the desired real form of \({\mathfrak g}\) [129].

6.5.5 Illustrations: \({\mathfrak {sl}}(4,\,{\mathbb R})\) versus \({\mathfrak {sl}}(2,\,{\mathbb H})\)

We shall exemplify the reconstruction of real algebras from Vogan diagrams by considering two examples of real forms of \({\mathfrak {sl}}(4,\,{\mathbb C})\). The diagrams are shown in Figure 35.
Figure 35

The Vogan diagrams associated to a \({\mathfrak {sl}}(4{\mathbb R})\) and \({\mathfrak {sl}}(2{\mathbb H})\) subalgebra.

The θ involutions they describe are (the upper signs correspond to the left-hand side diagram, the lower signs to the right-hand side diagram):
$$\begin{array}{*{20}c} {\theta {H_{{\alpha _1}}}} & = & {{H_{{\alpha _3}}},} & {{H_{{\alpha _2}}}} & = & {{H_{{\alpha _2}}},} & {\theta {H_{{\alpha _3}}}} & = & {{H_{{\alpha _1}}}}\\ {\theta {E_{{\alpha _1}}}} & = & {{E_{{\alpha _3}}},} & {\theta {E_{{\alpha _2}}}} & = & {\mp {E_{{\alpha _2}}},} & {\theta {E_{{\alpha _3}}}} & = & {{E_{{\alpha _1}}}{.}}\end{array}$$
(6.106)
Using the commutations relations
$$\begin{array}{*{20}c} {\left[ {{E_{{\alpha _1}}},{E_{{\alpha _2}}}} \right] = {E_{{\alpha _1} + {\alpha _2}}},\quad \quad \quad \quad \quad \;\;}\\ {\left[ {{E_{{\alpha _2}}},{E_{{\alpha _3}}}} \right] = {E_{{\alpha _2} + {\alpha _3}}},\quad \quad \quad \quad \quad \;\;}\\ {\left[ {{E_{{\alpha _1} + {\alpha _2}}},{E_{{\alpha _3}}}} \right] = {E_{{\alpha _1} + {\alpha _2} + {\alpha _3}}} = \left[ {{E_{{\alpha _1}}},{E_{{\alpha _2} + {\alpha _3}}}} \right]}\end{array}$$
(6.107)
we obtain
$$\theta {E_{{\alpha _1} + {\alpha _2}}} = \pm {E_{{\alpha _2} + {\alpha _3}}},\qquad \theta {E_{{\alpha _2} + {\alpha _3}}} = \pm {E_{{\alpha _1} + {\alpha _2}}},\qquad \theta {E_{{\alpha _1} + {\alpha _2} + {\alpha _3}}} = \mp {E_{{\alpha _1} + {\alpha _2} + {\alpha _3}}}{.}$$
(6.108)
Let us consider the left-hand side diagram. The corresponding +1 θ-eigenspace \({\mathfrak k}\) has the following realisation,
$${\mathfrak{k}} = {\rm{span}}\left\{{{H_{{\alpha _1}}} + {H_{{\alpha _3}}},\,{H_{{\alpha _2}}},\,{E_{{\alpha _1}}} + {E_{{\alpha _3}}},\,{E_{- {\alpha _1}}} + {E_{- {\alpha _3}}},\,{E_{{\alpha _1} + {\alpha _2}}} + {E_{{\alpha _2} + \alpha 3}},\,{E_{- {\alpha _1} - {\alpha _2}}} + {E_{- {\alpha _2} - \alpha 3}}} \right\},$$
(6.109)
and the −1 θ-eigenspace \({\mathfrak p}\) is given by
$$\begin{array}{*{20}c} {{\mathfrak{p}} = {\rm{span}}\{{H_{{\alpha _1}}} - {H_{{\alpha _3}}},\,{E_{\pm {\alpha _2}}},\,{E_{{\alpha _1}}} - {E_{{\alpha _3}}},\,{E_{- {\alpha _1}}} - {E_{- {\alpha _3}}},\quad \quad \quad \quad \quad}\\ {\quad {E_{{\alpha _1} + {\alpha _2}}} - {E_{{\alpha _2} + {\alpha _3}}},{E_{- {\alpha _1} - {\alpha _2}}} - {E_{- {\alpha _2} - {\alpha _3}}},{E_{\pm ({\alpha _1} + {\alpha _2} + {\alpha _3})}}\} {.}}\end{array}$$
(6.110)
The intersection \({{\mathfrak c}_0} \cap {\mathfrak k}\) then leads to the \({\mathfrak {so}}(4,\,{\mathbb R}) = {\mathfrak {so}}(3,\,{\mathbb R}) \oplus {\mathfrak {so}}(3,\,{\mathbb R})\) algebra
$$\begin{array}{*{20}c} {{{\mathfrak{k}}_0} = {\rm{span}}\{i({H_{{\alpha _1}}} + {H_{{\alpha _3}}}), ({E_{{\alpha _1}}} + {E_{{\alpha _3}}} - {E_{- {\alpha _1}}} - {E_{- {\alpha _3}}}), i({E_{{\alpha _1}}} + {E_{{\alpha _3}}} + {E_{- {\alpha _1}}} + {E_{- {\alpha _3}}})\}}\\ {\oplus\;{\rm{span}}\{i({H_{{\alpha _1}}} + 2 {H_{{\alpha _2}}} + {H_{{\alpha _3}}}), ({E_{{\alpha _1} + {\alpha _2}}} + {E_{{\alpha _2} + {\alpha _3}}} - {E_{- ({\alpha _1} + {\alpha _2})}} - {E_{- ({\alpha _2} + {\alpha _3})}}),}\\ {i({E_{{\alpha _1} + {\alpha _2}}} + {E_{{\alpha _2} + {\alpha _3}}} + {E_{- ({\alpha _1} + {\alpha _2})}} + {E_{- ({\alpha _2} + {\alpha _3})}})\} ,\quad \quad \quad \quad \;\;}\end{array}$$
(6.111)
and the remaining noncompact generator subspace \({{\mathfrak p}_0} = i({{\mathfrak c}_0} \cap {\mathfrak p})\) becomes
$$\begin{array}{*{20}c} {{{\mathfrak{p}}_0} = {\rm{span}}\{{H_{{\alpha _1}}} - {H_{{\alpha _3}}},({E_{{\alpha _1}}} - {E_{{\alpha _3}}} + {E_{- {\alpha _1}}} - {E_{- {\alpha _3}}}),i({E_{{\alpha _1}}} - {E_{{\alpha _3}}} - {E_{- {\alpha _1}}} + {E_{- {\alpha _3}}}),\quad \quad}\\ {({E_{{\alpha _1} + {\alpha _2}}} - {E_{{\alpha _2} + {\alpha _3}}} + {E_{- ({\alpha _1} + {\alpha _2})}} - {E_{- ({\alpha _2} + {\alpha _3})}}),\quad \quad \quad \quad \quad \quad \quad}\\ {\quad \quad \quad i({E_{{\alpha _1} + {\alpha _2}}} - {E_{{\alpha _2} + {\alpha _3}}} - {E_{- ({\alpha _1} + {\alpha _2})}} + {E_{- ({\alpha _2} + {\alpha _3})}}),{E_{{\alpha _2}}} + {E_{- {\alpha _2}}},i({E_{{\alpha _2}}} - {E_{- {\alpha _2}}}),}\\ {{E_{{\alpha _1} + {\alpha _2} + {\alpha _3}}} + {E_{- ({\alpha _1} + {\alpha _2} + {\alpha _3})}},i({E_{{\alpha _1} + {\alpha _2} + {\alpha _3}}} - {E_{- ({\alpha _1} + {\alpha _2} + {\alpha _3})}})\} {.}\quad \quad \quad \;}\end{array}$$
(6.112)
Doing the same exercise for the second diagram, we obtain the real algebra \({\mathfrak {sl}}(2,\,{\mathbb H})\) with \({{\mathfrak k}_0} = {\mathfrak {so}}(5,\,{\mathbb R}) = {\mathfrak {sp}}(4,\,{\mathbb R})\), which is a 10-parameter compact subalgebra, and \({{\mathfrak p}_0}\) given by
$$\begin{array}{*{20}c} {{{\mathfrak{p}}_0} = {\rm{span}}\{{H_{{\alpha _1}}} - {H_{{\alpha _3}}},({E_{{\alpha _1}}} - {E_{{\alpha _3}}} + {E_{- {\alpha _1}}} - {E_{- {\alpha _3}}}),i({E_{{\alpha _1}}} - {E_{{\alpha _3}}} - {E_{- {\alpha _1}}} + {E_{- {\alpha _3}}}),}\\ {({E_{{\alpha _1} + {\alpha _2}}} + {E_{{\alpha _2} + {\alpha _3}}} + {E_{- ({\alpha _1} + {\alpha _2})}} + {E_{- ({\alpha _2} + {\alpha _3})}}),\quad \quad \quad \quad \quad}\\ {i({E_{{\alpha _1} + {\alpha _2}}} + {E_{{\alpha _2} + {\alpha _3}}} - {E_{- ({\alpha _1} + {\alpha _2})}} - {E_{- ({\alpha _2} + {\alpha _3})}})\} {.}\quad \quad \quad \quad} \end{array}$$
(6.113)

6.5.6 A pictorial summary — All real simple Lie algebras (Vogan diagrams)

The following tables provide all real simple Lie algebras and the corresponding Vogan diagrams. The restrictions imposed on some of the Lie algebra parameters eliminate the consideration of isomorphic algebras. See [129] for the derivation.
Table 16

Vogan diagrams (A n series)

A n series, n ≥ 1

Vogan diagram

Maximal compact subalgebra

\({\mathfrak s}{\mathfrak u}(n + 1)\)

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\({\mathfrak s}{\mathfrak u}(n + 1)\)

\({\mathfrak s}{\mathfrak u}(p,\,q)\)

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\({\mathfrak s}{\mathfrak u}(p)\, \oplus \,{\mathfrak s}{\mathfrak u}(q)\, \oplus \,u(1)\)

\({\mathfrak s}{\mathfrak l}(2n,\,{\mathbb R})\)

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\({\mathfrak s}{\mathfrak o}(2n)\)

\({\mathfrak s}{\mathfrak l}(2n + 1,\,{\mathbb R})\)

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\({\mathfrak s}{\mathfrak o}(2n + 1)\)

\({\mathfrak s}{\mathfrak l}(n + 1,\,{\mathbb H})\)

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\({\mathfrak s}{\mathfrak p}(n + 1)\)

Table 17

Vogan diagrams (B n series)

B n series, n ≥ 2

Vogan diagram

Maximal compact subalgebra

\({\mathfrak s}{\mathfrak o}(2n + 1)\)

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\({\mathfrak s}{\mathfrak o}(2n + 1)\)

\(\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak o}(p,\,q)} \\ {p \leq n - {1 \over 2},\,q = 2n + 1 - p} \\ \end{array}\)

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\({\mathfrak s}{\mathfrak o}(p)\, \oplus \,{\mathfrak s}{\mathfrak o}(q)\)

Table 18

Vogan diagrams (C n series)

C n series, n > 3

Vogan diagram

Maximal compact subalgebra

\({\mathfrak s}{\mathfrak p}(n)\)

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\({\mathfrak s}{\mathfrak p}(n)\)

\(\begin{array}{*{20}c}{{\mathfrak s}{\mathfrak p}(p,\,q)} \\ {0 < p \leq {n \over 2},\,q = n - p} \\ \end{array}\)

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\({{\mathfrak s}{\mathfrak p}(p)\, \oplus \,{\mathfrak s}{\mathfrak p}(q)}\)

\({{\mathfrak s}{\mathfrak p}(n,\,{\mathbb R})}\)

\({{\mathfrak u}(n)}\)

Table 19

Vogan diagrams (D n series)

D n series, n ≥ 4

Vogan diagram

Maximal compact subalgebra

\({\mathfrak s}{\mathfrak o}(2n)\)

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\({\mathfrak s}{\mathfrak o}(2n)\)

\(\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak o}(2p,\,2q)} \\ {0 < p \leq {n \over 2},\,q = n - p} \\ \end{array}\)

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\({{\mathfrak s}{\mathfrak o}(2p)\, \oplus \,{\mathfrak s}{\mathfrak o}(2q)}\)

\({{\mathfrak s}{\mathfrak o}\ast (2n)}\)

\({{\mathfrak u}(n)}\)

\(\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak o}(2p + 1,\,2q + 1)} \\ {0 < p \leq {{n - 1} \over 2},} \\ {q = n - p - 1} \\ \end{array}\)

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\({\mathfrak s}{\mathfrak o}(2p + 1)\, \oplus \,{\mathfrak s}{\mathfrak o}(2q + 1)\)

Table 20

Vogan diagrams (G2 series)

G 2

Vogan diagram

Maximal compact subalgebra

G 2

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G 2

G 2(2)

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\({\mathfrak s}{\mathfrak u}(2)\, \oplus \,{\mathfrak s}{\mathfrak u}(2)\)

Table 21

Vogan diagrams (F4 series)

F4 series

Vogan diagram

Maximal compact subalgebra

F 4

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F 4

F 4(4)

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\({\mathfrak s}{\mathfrak p}(3)\, \oplus \,{\mathfrak s}{\mathfrak u}(2)\)

F4(−20)

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\({\mathfrak s}{\mathfrak o}(9)\)

Table 22

Vogan diagrams (E6 series)

E 6

Vogan diagram

Maximal compact subalgebra

E 6

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E 6

E 6(6)

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\({\mathfrak s}{\mathfrak p}(4)\)

E 6(2)

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\({\mathfrak s}{\mathfrak u}(6)\, \oplus \,{\mathfrak s}{\mathfrak u}(2)\)

E 6(−14)

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\({\mathfrak s}{\mathfrak u}(10)\, \oplus \,{\mathfrak u}(1)\)

E 6(−26)

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F 4

Table 23

Vogan diagrams (E7 series)

E 7

Vogan diagram

Maximal compact subalgebra

E 7

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E 7

E 7(7)

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\({\mathfrak s}{\mathfrak u}(8)\)

E 7(43)

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\({\mathfrak s}{\mathfrak o}(12)\, \oplus \,{\mathfrak s}{\mathfrak u}(2)\)

E 7( 25)

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\({E_6}\, \oplus {\mathfrak u}(1)\)

Table 24

Vogan diagrams (E8 series)

E 8

Vogan diagram

Maximal compact subalgebra

E 8

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E 8

E 8(8)

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\({\mathfrak s}{\mathfrak o}(16)\)

E 8( 24)

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\({E_7}\, \oplus {\mathfrak s}{\mathfrak u}(2)\)

Using these diagrams, the matrix I p, q defined by Equation (6.93), and the three matrices
$${J_n} = \left({\begin{array}{*{20}c} 0 &{I{d^{n \times n}}}\\ {- I{d^{n \times n}}}&0\end{array}} \right),$$
(6.114)
$${K_{p,q}} = \left({\begin{array}{*{20}c} {I{d^{p \times p}}} & 0 & 0 & 0\\ 0 & {- I{d^{q \times q}}} & 0 & 0\\ 0 & 0 & {I{d^{p \times p}}} & 0\\ 0 & 0 & 0 & {- I{d^{q \times q}}} \end{array}} \right),$$
(6.115)
$${L_{p,q}} = {K_{p,q}}{J_{p + q}} = \left({\begin{array}{*{20}c} 0 & 0 & {I{d^{p \times p}}} & 0\\ 0 & 0 & 0 & {- I{d^{q \times q}}}\\ {- I{d^{p \times p}}} & 0 & 0 & 0\\ 0 & {I{d^{q \times q}}} & 0 & 0 \end{array}} \right),$$
(6.116)
we may check that the involutive automorphisms of the classical Lie algebras are all conjugate to one of the types listed in Table 25.
Table 25

List of all involutive automorphisms (up to conjugation) of the classical compact real Lie algebras [93]. The first column gives the complexification \({\mathfrak u}_0^{\mathbb C}\) of the compact real algebra \({\mathfrak u_0}\), the second \({\mathfrak u_0}\), the third the involution τ that \({\mathfrak u_0}\) defines in \({\mathfrak u^\mathbb C}\), and the fourth a non-compact real subalgebra \({\mathfrak g_0}\) of \({\mathfrak u^\mathbb C}\) aligned with the compact one. In the second table, the second column displays the involution that \({\mathfrak g_0}\) defines on \({\mathfrak u^\mathbb C}\), the third the involutive automorphism of \({\mathfrak u_0}\), i.e., the Cartan conjugation θ = στ, and the last column indicates the common compact subalgebra \({\mathfrak k_0}\) of \({\mathfrak u_0} = {\mathfrak k_0}\, \oplus \,i\,{\mathfrak p_0}\) and \({\mathfrak g_0} = {\mathfrak k_0}\, \oplus \,{\mathfrak p_0}\).

\({\mathfrak u^\mathbb C}\)

\({\mathfrak u_0}\)

τ

\({\mathfrak g_0}\)

\({\mathfrak s}{\mathfrak l}(n,\,{\mathbb C})\)

\({\mathfrak s}{\mathfrak u}(n)\)

X

A I \({\mathfrak s}{\mathfrak l}(n,\,{\mathbb R})\)

\({\mathfrak s}{\mathfrak l}(2n,\,{\mathbb C})\)

\({\mathfrak s}{\mathfrak u}(2n)\)

X

A II \({\mathfrak s}{\mathfrak u}\ast (2n)\)

\({\mathfrak s}{\mathfrak l}(p + q,\,{\mathbb C})\)

\({\mathfrak s}{\mathfrak u}(p + q)\)

X

A III \({\mathfrak s}{\mathfrak u}(p,\,q)\)

\({\mathfrak s}{\mathfrak o}(p + q,\,{\mathbb C})\)

\({\mathfrak s}{\mathfrak o}(p + q,\,{\mathbb R})\)

\(\overline X\)

B I, D I \({\mathfrak s}{\mathfrak o}(p,\,q)\)

\({\mathfrak s}{\mathfrak o}(2n,\,{\mathbb C})\)

\({\mathfrak s}{\mathfrak o}(2n,\,{\mathbb R})\)

\(\overline X\)

D III \({\mathfrak s}{\mathfrak o}\ast (2n)\)

\({\mathfrak s}{\mathfrak p}(n,\,{\mathbb C})\)

\({\mathfrak u}{\mathfrak s}{\mathfrak p}(n)\)

\(- {J_n}\overline X \,{J_n}\)

C I \({\mathfrak s}{\mathfrak p}(n,\,{\mathbb R})\)

\({\mathfrak s}{\mathfrak p}(p + q,\,{\mathbb C})\)

\({\mathfrak u}{\mathfrak s}{\mathfrak p}(p + q)\)

\(- {J_{p + q}}\overline X {J_{p + q}}\)

C III \({\mathfrak s}{\mathfrak p}(p,\,q)\)

\({\mathfrak u^\mathbb C}\)

σ

θ

\({\mathfrak k_0}\)

\({\mathfrak s}{\mathfrak l}(n,\,{\mathbb C})\)

\(\overline X\)

X t

\({\mathfrak s}{\mathfrak o}(n,\,{\mathbb R})\)

\({\mathfrak s}{\mathfrak l}(2n,\,{\mathbb C})\)

\(- {J_n}\overline X \,{J_n}\)

J n X t J n

\({\mathfrak u}{\mathfrak s}{\mathfrak p}(2n)\)

\({\mathfrak s}{\mathfrak l}(p + q,\,{\mathbb C})\)

Ip,q X Ip,q

I p,q X I p,q

\({\mathfrak s}{\mathfrak o}(n,\,{\mathbb R})\)

\({\mathfrak s}{\mathfrak o}(p + q,\,{\mathbb C})\)

\({I_{p,q}}\overline X {I_{p,q}}\)

I p,q X I p,q

\({\mathfrak s}{\mathfrak o}(p,\,R)\, \oplus \,{\mathfrak s}{\mathfrak o}(q,\,R)\)

\({\mathfrak s}{\mathfrak o}(2n,\,{\mathbb C})\)

\(- {J_n}\overline X {J_n}\)

J n X J n

\({\mathfrak s}{\mathfrak u}(n)\, \oplus \,{\mathfrak u}(1)\)

\({\mathfrak s}{\mathfrak p}(n,\,{\mathbb C})\)

\(\overline X\)

J n X J n

\({\mathfrak s}{\mathfrak u}(n)\, \oplus \,{\mathfrak u}(1)\)

\({\mathfrak s}{\mathfrak p}(p + q,\,{\mathbb C})\)

Kp,q X Kp,q

L p,q X t L p,q

\({\mathfrak s}{\mathfrak p}(p)\, \oplus \,{\mathfrak s}{\mathfrak p}(q)\)

For completeness we remind the reader of the definitions of matrix algebras (\({\mathfrak {su}}(p,\,q)\) has been defined in Equation (6.93)):
$$\begin{array}{*{20}c} {{{\mathfrak{su}}^ \ast}(2n) = \left\{{X\left\vert {X{J_n} - {J_n}\bar X = 0,{\rm{Tr}}X = 0,X \in {{\mathbb{C}}^{2n \times 2n}}} \right.} \right\}}\\ {= \left\{{\left({\begin{array}{*{20}c} A & C\\ {- \bar C} & {\bar A}\end{array}} \right)\left\vert {\begin{array}{*{20}c} {A,C \in {{\mathbb{C}}^{n \times n}}}\\ {{\rm{Re}}\left[ {{\rm{Tr}}A} \right] = 0}\end{array}} \right.} \right\},}\quad\quad \end{array}$$
(6.117)
$$\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak o}(p,\,q) = \left\{{X\left\vert {X{I_{p,q}} + {I_{p,q}}{X^t} = 0,\,X = - {X^t},\,X \in {{\mathbb R}^{(p + q) \times (p + q)}}} \right.} \right\}\quad \quad} \\ {= \left\{{\left({\begin{array}{*{20}c} A & C \\ {{C^t}} & B \\ \end{array}} \right)\left\vert {\begin{array}{*{20}c} {A = - {A^t} \in {{\mathbb R}^{p \times p}},\,B = - {B^t} \in {{\mathbb R}^{q \times q}},} \\ {C \in {{\mathbb R}^{p \times q}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \;} \\ \end{array}} \right.} \right\},} \\ \end{array}$$
(6.118)
$$\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak o}^{\ast}(2n) = \left\{{X\left\vert {{X^t}{J_n} + {J_n}\bar X = 0,\,X = - {X^t},\,X \in {{\mathbb C}^{2n \times 2n}}} \right.} \right\}\;\;\;\,} \\ {= \left\{{\left({\begin{array}{*{20}c} A & B \\ {- \bar B} & {\bar A} \\ \end{array}} \right)\left\vert {A = - {A^t},\,B = {B^\dagger} \in {{\mathbb C}^{n \times n}}} \right.} \right\},} \\ \end{array}$$
(6.119)
$$\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak p}(n,\;\,{\mathbb R}) = \left\{{X\left\vert {{X^t}{J_n} + {J_n}X = 0,\,{\rm{Tr}}\;X = 0,\;X \in {{\mathbb R}^{2n \times 2n}}} \right.} \right\}\quad \quad \quad} \\ {= \left\{{\left({\begin{array}{*{20}c} A & B \\ C & {- {A^t}} \\ \end{array}} \right)\left\vert {A,\,B = {B^t},\;\;C = {C^t} \in {{\mathbb R}^{n \times n}}} \right.} \right\}\;,} \\ \end{array}$$
(6.120)
$$\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak p}(n,\;{\mathbb C})) = \left\{{X\left\vert {{X^t}{J_n} + {J_n}X = 0,{\rm{Tr}}\;X = 0,\;X \in {{\mathbb C}^{2n \times 2n}}} \right.} \right\}\quad \quad} \\ {= \left\{{\left({\begin{array}{*{20}c} A & B \\ C & {- {A^t}} \\ \end{array}} \right)\left\vert {A,\,B = {B^t},\,C = {C^t} \in {{\mathbb C}^{n \times n}}} \right.} \right\}\;,} \\ \end{array}$$
(6.121)
$$\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak p}(p,\,q) = \left\{{X\left\vert {{X^t}{K_{p,q}} + {K_{p,q}}\bar X = 0,\,{\rm{Tr}}\;X = 0,\,\;X \in {{\mathbb C}^{(p + q) \times (p + q)}}} \right.} \right\}\quad \quad \;\,} \\ {= \left\{{\left({\begin{array}{*{20}c} A & P & Q & R \\ {{P^\dagger}} & B & {{R^t}} & S \\ {- \bar Q} & {\bar R} & {\bar A} & {- \bar P} \\ {{R^\dagger}} & {- \bar S} & {- {P^t}} & {\bar B} \\ \end{array}} \right)\left\vert {\begin{array}{*{20}c} {A,\,Q \in {{\mathbb C}^{p \times p}}} \\ {P,\,R \in {{\mathbb C}^{p \times q}},\,S \in {{\mathbb C}^{q \times p}}} \\ {A = - {A^\dagger},\,B = - {B^\dagger}} \\ {Q = {Q^t},\,S = {S^t}} \\ \end{array}} \right.} \right\}\;,} \\ \end{array}$$
(6.122)
$${\mathfrak u}{\mathfrak s}{\mathfrak p}(2p,\,2q) = {\mathfrak s}{\mathfrak u}(2p,\,2q) \cap {\mathfrak s}{\mathfrak p}(2p + 2q).$$
(6.123)
Alternative definitions are:
$$\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak p}(p,q) = \{X \in {\mathfrak g}{\mathfrak l}(p + q,{\mathbb H})\vert \bar X\,{I_{p,q}} + {I_{p,q}}\,X = 0\} ,\;\;\,} \\ {{\mathfrak s}{\mathfrak p}(n,{\mathbb R}) = \{X \in {\mathfrak g}{\mathfrak l}(2\,n,{\mathbb R})\vert {X^t}\,{J_n} + {J_n}\,X = 0\} ,\quad \quad \;\;} \\ {{\mathfrak s}{\mathfrak l}(n,{\mathbb H}) = \{X \in {\mathfrak g}{\mathfrak l}(n,{\mathbb H})\vert \bar X + X = 0\} ,\quad \quad \quad \quad \quad \;\,} \\ {{{\mathfrak s}{\mathfrak o}^{\ast}}(2\,n) = \{X \in {\mathfrak s}{\mathfrak u}(n,n)\vert {X^t}\,{K_n} + {K_n}\,X = 0\} .\quad \quad \;\;\;\,} \\ \end{array}$$
(6.124)
For small dimensions we have the following isomorphisms:
$$\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak u}(1,2) \simeq {\mathfrak s}{\mathfrak u}(1,1) \simeq {\mathfrak s}{\mathfrak u}(1,{\mathbb R}) \simeq {\mathfrak s}{\mathfrak u}(2,{\mathbb R}),\,} \\ {{\mathfrak s}{\mathfrak u}(2,{\mathbb C}) \simeq {\mathfrak s}{\mathfrak u}(1,3),\quad \quad \quad \quad \quad \quad \quad \quad \;\;\;} \\ {{{\mathfrak s}{\mathfrak u}^{\ast}}(4) \simeq {\mathfrak s}{\mathfrak u}(2) \oplus {\mathfrak s}{\mathfrak u}(1,1),\quad \quad \quad \quad \quad} \\ {{{\mathfrak s}{\mathfrak u}^{\ast}}(6) \simeq {\mathfrak s}{\mathfrak u}(3,1),\quad \quad \quad \quad \quad \quad \quad \quad \,} \\ {{\mathfrak s}{\mathfrak u}(1,1) \simeq {\mathfrak s}{\mathfrak u}(1,4),\quad \quad \quad \quad \quad \quad \quad \quad \;} \\ {{\mathfrak s}{\mathfrak u}(2,{\mathbb H}) \simeq {\mathfrak s}{\mathfrak u}(1,5),\quad \quad \quad \quad \quad \quad \quad \quad \;\;\;} \\ {{\mathfrak s}{\mathfrak u}(2,2) \simeq {\mathfrak s}{\mathfrak u}(2,4),\quad \quad \quad \quad \quad \quad \quad \quad \;\,} \\ {{\mathfrak s}{\mathfrak u}(4,{\mathbb R}) \simeq {\mathfrak s}{\mathfrak u}(3,3),\quad \quad \quad \quad \quad \quad \quad \quad \;\;\,} \\ {{{\mathfrak s}{\mathfrak u}^{\ast}}(8) \simeq {\mathfrak s}{\mathfrak u}(2,6).\quad \quad \quad \quad \quad \quad \quad \quad \,} \\ \end{array}$$
(6.125)

6.6 Tits-Satake diagrams

The classification of real forms of a semi-simple Lie algebra, using Vogan diagrams, rests on the construction of a maximally compact Cartan subalgebra. On the other hand, the Iwasawa decomposition emphasizes the role of a maximally noncompact Cartan subalgebra. The consideration of these Cartan subalgebras leads to another way to classify real forms of semi-simple Lie algebras, developed mainly by Araki [5], and based on so-called Tits-Satake diagrams [161, 155].

6.6.1 Example 1: \({\mathfrak {su}}(3,\,2)\)

6.6.1.1 Diagonal description
At the end of Section 6.5.3, we obtained a matrix representation of a maximally noncompact Cartan subalgebra of \({\mathfrak {su}}(3,\,2)\) in terms of the natural description of this algebra. To facilitate the forthcoming discussion, we find it useful to use an equivalent description, in which the matrices representing this Cartan subalgebra are diagonal, as this subalgebra will now play a central role. It is obtained by performing a similarity transformation XS T X S, where
$$S = \left({\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 \\ 0 & {{1 \over {\sqrt 2}}} & 0 & 0 & {{1 \over {\sqrt 2}}} \\ 0 & 0 & {{1 \over {\sqrt 2}}} & {{1 \over {\sqrt 2}}} & 0 \\ 0 & 0 & {- {1 \over {\sqrt 2}}} & {{1 \over {\sqrt 2}}} & 0 \\ 0 & {- {1 \over {\sqrt 2}}} & 0 & 0 & {{1 \over {\sqrt 2}}} \\ \end{array}} \right)\;\,.$$
(6.126)
In this new “diagonal” description, the conjugation σ (see Equation (6.94)) becomes
$$\sigma (X) = - {\tilde I_{3,2}}{X^\dagger}{\tilde I_{3,2}},$$
(6.127)
where
$${\tilde I_{3,2}} = {S^T}{I_{3,2}}S = \left({\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array}} \right)\;\,.$$
(6.128)
The Cartan involution has the following realisation:
$$\theta (X) = {\tilde I_{3,2}}X\,{\tilde I_{3,2}}.$$
(6.129)
In terms of the four matrices introduced in Equation (6.84), the generators defining this Cartan subalgebra \({\mathfrak h}\) reads
$$\begin{array}{*{20}c} {{h_1} = {H_3},\quad \quad \quad \quad \quad \quad \,} & {{h_2} = {H_2} + {H_3} + {H_4},\quad \quad \quad \;\;} \\ {{h_3} = i(2\,{H_1} + 2\,{H_2} + {H_3}),} & {{h_4} = i(2\,{H_1} + {H_2} + {H_3} + {H_4}).} \\ \end{array}$$
(6.130)
Let us emphasize that we have numbered the basis generators of \({\mathfrak h} = {\mathfrak a} \oplus {\mathfrak t}\) by first choosing those in \({\mathfrak a}\), then those in \({\mathfrak t}\).
6.6.1.2 Cartan involution and roots

The standard matrix representation of \({\mathfrak {su}}(5)\) constitutes a compact real Lie subalgebra of \({\mathfrak {sl}}(5,\,{\mathbb C})\) aligned with the diagonal description of the real form \({\mathfrak {su}}(3,\, 2)\). Moreover, its Cartan subalgebra \({{\mathfrak h}_1}\) generated by purely imaginary combinations of the four diagonal matrices H k is such that its complexification \({{\mathfrak h}^{\mathbb C}}\) contains \({\mathfrak h}\). Accordingly, the roots it defines act both on \({{\mathfrak h}_0}\) and \({\mathfrak h}\). Note that on \({\mathfrak h}_{\mathbb R} = i\, {\mathfrak h}_0\), the roots take only real values.

Our first task is to compute the action of the Cartan involution θ on the root lattice. With this aim in view, we introduce two distinct bases on \({\mathfrak h}_{\mathbb R}^{\ast}\). The first one is {F1, F2, F3, F4}, which is dual to the basis {H1, H2, H3, H4} and is adapted to the relation \({\mathfrak h}_{\mathbb R} = i\, {\mathfrak h}_0\) The second one is {f1, f2, f3, f4}, dual to the basis {h1, h2, −ih3, −ih4}, which is adapted to the decomposition \({{\mathfrak h}_{\mathbb R}} = {\mathfrak a} \oplus i{\mathfrak t}\). The Cartan involution acts on these root space bases as
$$\theta \{{f^1},{f^2},{f^3},{f^4}\} = \{- {f^1}, - {f^2},{f^3},{f^4}\} .$$
(6.131)
From the relations (6.130) it is easy to obtain the expression of the {F k } (k =1, ⋯, 4) in terms of the {f k } and thus also the expressions for the simple roots α1 = 2F1F2, α2 = −F1 + 2F2F3, α3 = −F2 + 2F3F4 and α4 = −F3 + 2F4, defined by \({\mathfrak {h}_0}\),
$$\begin{array}{*{20}c} {{\alpha _1} = - {f^2} + 2\,{f^3} + 3\,{f^4},\quad} \\ {{\alpha _2} = - {f^1} + {f^2} + {f^3} - {f^4},\,} \\ {{\alpha _3} = 2\,{f^1},\quad \quad \quad \quad \quad \quad \;\,} \\ {{\alpha _4} = - {f^1} + {f^2} - {f^3} + {f^4}.\,} \\ \end{array}$$
(6.132)
It is then straightforward to obtain the action of θ on the roots, which, when expressed in terms of the \({\mathfrak {h}_0}\) simple roots themselves, is given by
$$\begin{array}{*{20}c} {\theta [{\alpha _1}] = {\alpha _1} + {\alpha _2} + {\alpha _3} + {\alpha _4},} \\ {\theta [{\alpha _2}] = - {\alpha _4},\quad \quad \quad \quad \quad \;\;} \\ {\theta [{\alpha _3}] = - {\alpha _3},\quad \quad \quad \quad \quad \;\;} \\ {\theta [{\alpha _4}] = - {\alpha _2}.\quad \quad \quad \quad \quad \;\;\,} \\ \end{array}$$
(6.133)
We see that the root α3 is real while α1, α2 and α4 are complex As a check of these results, we may, for instance, verify that
$$\theta {E_{{\alpha _1}}} = {\tilde I_{3,2}}\,K_2^1\,{\tilde I_{3,2}} = K_5^1 = {E_{{\alpha _1} + {\alpha _2} + {\alpha _3} + {\alpha _4}}}.$$
(6.134)
In fact, this kind of computation provides a simpler way to obtain Equation (6.133)
The basis {f1, f2, f3, f4} allows to define a different ordering on the root lattice, merely by considering the corresponding lexicographic order. In terms of this new ordering we obtain for instance α1 < 0 since the first nonzero component of α1 (in this case −1 along f2) is strictly negative. Similarly, one finds α2 < 0, α3 > 0, α4 < 0, α1 + α2 < 0, α2 + α3 > 0, α3 + α4 > 0, α1 + α2 + α3 > 0, α2 + α3 + α4 > 0, α1 + α2 + α3 + α4 > 0. A basis of simple roots, according to this ordering, is given by
$$\begin{array}{*{20}c} {{{\tilde \alpha}_1} = - {\alpha _4} = {f^1} - {f^2} + {f^3} - {f^4},\quad \quad \quad \quad \;\;} \\ {{{\tilde \alpha}_2} = {\alpha _1} + {\alpha _2} + {\alpha _3} + {\alpha _4} = {f^2} + 2\,{f^3} + 3\,{f^4},} \\ {{{\tilde \alpha}_3} = - {\alpha _1} = {f^2} - 2\,{f^3} - 3\,{f^4},\quad \quad \quad \quad \quad \;\;\,} \\ {{{\tilde \alpha}_4} = - {\alpha _2} = {f^1} - {f^2} - {f^3} + {f^4}.\quad \quad \quad \quad \;\;\,} \\ \end{array}$$
(6.135)
(We have put \({{\tilde \alpha}_4}\) in fourth position, rather than in second, to follow usual conventions.) The action of θ on this basis reads
$$\theta [{\tilde \alpha _1}] = - {\tilde \alpha _4},\qquad \theta [{\tilde \alpha _2}] = - {\tilde \alpha _3},\qquad \theta [{\tilde \alpha _3}] = - {\tilde \alpha _2},\qquad \theta [{\tilde \alpha _4}] = - {\tilde \alpha _1}.$$
(6.136)
These new simple roots are now all complex.
6.6.1.3 Restricted roots
The restricted roots are obtained by considering only the action of the roots on the noncompact Cartan generators h1 and h2. The two-dimensional vector space spanned by the restricted roots can be identified with the subspace spanned by f1 and f2; one simply projects out f3 and f4. In the notations β1 = f1f2 and β2 = f2, one gets as positive restricted roots:
$${\beta _1},\qquad {\beta _2},\qquad {\beta _1} + {\beta _2},\qquad {\beta _1} + 2{\beta _2},\qquad 2{\beta _2},\qquad 2({\beta _1} + {\beta _2}),$$
(6.137)
which are the positive roots of the (BC)2 (non-reduced) root system. The first four roots are degenerate twice, while the last two roots are nondegenerate. For instance, the two simple roots \({{\tilde \alpha}_1}\) and \({{\tilde \alpha}_4}\) project on the same restricted root β1, while the two simple roots \({{\tilde \alpha}_2}\) and \({{\tilde \alpha}_3}\) project on the same restricted root β2.

Counting multiplicities, there are ten restricted roots — the same number as the number of positive roots of \({\mathfrak {sl}}(5,\,{\mathbb C})\). No root of \({\mathfrak {sl}}(5,\,{\mathbb C})\) projects onto zero. The centralizer of \({\mathfrak a}\) consists only of \({\mathfrak a} \oplus {\mathfrak t}\).

6.6.2 Example 2: \({\mathfrak {su}}(4,\, 1)\)

6.6.2.1 Diagonal description
Let us now perform the same analysis within the framework of \({\mathfrak {su}}(4,\, 1)\). Starting from the natural description (6.92) of \({\mathfrak {su}}(4,\, 1)\), we first make a similarity transformation using the matrix
$$S = \left({\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & {{1 \over {\sqrt 2}}} & {{1 \over {\sqrt 2}}} \\ 0 & 0 & 0 & {- {1 \over {\sqrt 2}}} & {{1 \over {\sqrt 2}}} \\ \end{array}} \right)\;\,,$$
(6.138)
so that a maximally noncompact Cartan subalgebra can be taken to be diagonal and is explicitly given by
$${h_1} = {H_4},\qquad {h_2} = i\,{H_1},\qquad {h_3} = i\,{H_2},\qquad {h_4} = i(2\,{H_3} + {H_4}).$$
(6.139)
The corresponding \({\mathfrak {su}}(4,\, 1)\) in the \({\mathfrak {sl}}(5,\,{\mathbb C})\) algebra is still aligned with the natural matrix representation of \({\mathfrak {su}}(5)\). The Cartan involution is given by XĨ4,1 X Ĩ4,1 where Ĩ4,1 = S T I4,1 S. One has \({\mathfrak h}={\mathfrak a} \oplus {\mathfrak t}\) where the noncompact part \({\mathfrak a}\) is one-dimensional and spanned by h1, while the compact part t is three-dimensional and spanned by h2, h3 and h4.
6.6.2.2 Cartan involution and roots
In terms of the f i ’s, the standard simple roots now read
$$\begin{array}{*{20}c} {{\alpha _1} = 2\,{f^2} - {f^3},\quad \quad \quad \,} \\ {{\alpha _2} = - {f^2} + 2\,{f^3} - 2\,{f^4},} \\ {{\alpha _3} = - {f^1} - {f^2} + 3\,{f^4},\;\;} \\ {{\alpha _4} = 2\,{f^1}.\quad \quad \quad \quad \quad \;\,} \\ \end{array}$$
(6.140)
The Cartan involution acts as
$$\begin{array}{*{20}c} {\theta [{\alpha _1}] = {\alpha _1},\quad \;\;\;} \\ {\theta [{\alpha _2}] = {\alpha _2},\quad \;\;\;} \\ {\theta [{\alpha _3}] = {\alpha _3} + {\alpha _4},} \\ {\theta [{\alpha _4}] = - {\alpha _4},\quad \;} \\ \end{array}$$
(6.141)
showing that α1 and α2 are imaginary, α4 is real, while α3 is complex.
A calculation similar to the one just described above, using as ordering rules the lexicographic ordering defined by the dual of the basis in Equation (6.139), leads to the new system of simple roots,
$$\begin{array}{*{20}c} {{{\tilde \alpha}_1} = - {\alpha _1} - {\alpha _2} - {\alpha _3},} \\ {{{\tilde \alpha}_2} = {\alpha _1} + {\alpha _2},\quad \quad \;\;} \\ {{{\tilde \alpha}_3} = - {\alpha _2},\;\;\quad \quad \quad} \\ {{{\tilde \alpha}_4} = {\alpha _2} + {\alpha _3} + {\alpha _4},\;} \\ \end{array}$$
(6.142)
which transform as
$$\begin{array}{*{20}c} {\theta [{{\tilde \alpha}_1}] = - {{\tilde \alpha}_4} - {{\tilde \alpha}_2} - {{\tilde \alpha}_3},}\\ {\theta [{{\tilde \alpha}_2}] = {{\tilde \alpha}_2},\quad \quad \quad \quad \;\;}\\ {\theta [{{\tilde \alpha}_3}] = {{\tilde \alpha}_3},\quad \quad \quad \quad \;\;}\\ {\theta [{{\tilde \alpha}_4}] = - {{\tilde \alpha}_1} - {{\tilde \alpha}_2} - {{\tilde \alpha}_3}\;\;}\\ \end{array}$$
(6.143)
under the Cartan involution. Note that in this system, the simple roots \({{\tilde \alpha}_2}\) and \({{\tilde \alpha}_3}\) are imaginary and hence fixed by the Cartan involution, while the other two simple roots are complex.
6.6.2.3 Restricted roots
The restricted roots are obtained by considering the action of the roots on the single noncompact Cartan generator h1. The one-dimensional vector space spanned by the restricted roots can be identified with the subspace spanned by f1; one now simply projects out f2, f3 and f4. With the notation β1 = f1, we get as positive restricted roots
$${\beta _1},\qquad 2{\beta _1},$$
(6.144)
which are the positive roots of the (BC)1 (non-reduced) root system. The first root is six times degenerate, while the second one is nondegenerate. The simple roots \({{\tilde \alpha}_1}\) and \({{\tilde \alpha}_4}\) project on the same restricted root β1, while the imaginary root \({{\tilde \alpha}_2}\) and \({{\tilde \alpha}_3}\) project on zero (as does also the non-simple, positive, imaginary root \({{\tilde \alpha}_2} + {{\tilde \alpha}_3}\)).

Let us finally emphasize that the centralizer of \({\mathfrak a}\) in \({\mathfrak {su}}(4,\, 1)\) is now given by \({\mathfrak a} \oplus {\mathfrak m}\), where \({\mathfrak m}\) is the center of \({\mathfrak a}\) in \({\mathfrak k}\) (i.e., the subspace generated by the compact generators that commute with H4) and contains more than just the three compact Cartan generators h2, h3 and h4. In fact, m involves also the root vectors E β whose roots restrict to zero. Explicitly, expressed in the basis of Equation (6.85), these roots read β = ϵ p ϵ q with p, q = 1, 2, or 3 and are orthogonal to α4. The algebra m constitutes a rank 3, 9-dimensional Lie algebra, which can be identified with \({\mathfrak {su}}(3) \oplus {\mathfrak u}(1)\).

6.6.3 Tits-Satake diagrams: Definition

We may associate with each of the constructions of these simple root bases a Tits-Satake diagram as follows. We start with a Dynkin diagram of the complex Lie algebra and paint in black (●) the imaginary simple roots, i.e., the ones fixed by the Cartan involution. The others are represented by a white vertex (○). Moreover, some double arrows are introduced in the following way. It can be easily proven (see Section 6.6.4) that for real semi-simple Lie algebras, there always exists a basis of simple roots B that can be split into two subsets: B0 = {αr+1, …, α l } whose elements are fixed by θ (they correspond to the black vertices) and B B0 = {α1, …, α r } (corresponding to white vertices) such that
$$\forall {\alpha _k} \in B\backslash {B_0}:\theta [{\alpha _k}] = - {\alpha _{\pi (k)}} + \sum\limits_{j = r + 1}^l {a_k^j} \,{\alpha _j},$$
(6.145)
where π is an involutive permutation of the r indices of the elements of B B0. Accordingly, π contains cycles of one or two elements. In the Tits-Satake diagram, we connect by a double arrow all pairs of distinct simple roots α k and α π (k) in the same two-cycle orbit. For instance, for \({\mathfrak {su}}(3,\, 2)\) and \({\mathfrak {su}}(4,\, 1)\), we obtain the diagrams in Figure 36.
Figure 36

Tits-Satake diagrams for \({\mathfrak {su}}(3,\, 2)\) and \({\mathfrak {su}}(4,\, 1)\).

6.6.4 Formal considerations

Tits-Satake diagrams provide a lot of information about real semi-simple Lie algebras. For instance, we can read from them the full action of the Cartan involution as we now briefly pass to show. More information may be found in [5, 93].

The Cartan involution allows one to define a closed subsystem28 Δ0 of Δ:
$${\Delta _0} = \{\alpha \in \Delta \vert \theta [\alpha ] = \alpha \} ,$$
(6.146)
which is the system of imaginary roots. These project to zero when restricted to the maximally noncompact Cartan subalgebra. As we have seen in the examples, it is useful to use an ordering adapted to the Cartan involution. This can be obtained by considering a basis of \({\mathfrak h}\) constituted firstly by elements of \({\mathfrak a}\) followed by elements of \({\mathfrak t}\). If we use the lexicographic order defined by the dual of this basis, we obtain a root ordering such that if α ∉ Δ0 is positive, θ[α] is negative since the real part comes first and changes sign. Let B be a simple root basis built with respect to this ordering and let B0 = B ∩ Δ0. Then we have
$$B = \{{\alpha _1}, \ldots ,{\alpha _l}\} \qquad {\rm{and}}\qquad {B_0} = \{{\alpha _{r + 1}}, \ldots ,{\alpha _l}\}{.}$$
(6.147)
The subset B0 is a basis for Δ0. To see this, let B B0 = {α1, …, α r }. If \(\beta = \sum\nolimits_{k = 1}^l {{b^k}\,{\alpha _k}}\) is, say, a positive root (i.e., with coefficients b k ≥ 0) belonging to Δ0, then βθ[β] = 0 is given by a sum of positive roots, weighted by non-negative coefficients, \(\sum\nolimits_{k = 1}^r {{b^k}} ({\alpha _k} - \theta [{\alpha _k}])\). As a consequence, the coefficients b k are all zero for k = 1, ⋯, r and B0 constitutes a basis of Δ0, as claimed.
To determine completely θ we just need to know its action on a basis of simple roots. For those belonging to B0 it is the identity, while for the other ones we have to compute the coefficients \(a_k^j\), in Equation (6.145). These are obtained by solving the linear system given by the scalar products of these equations with the elements of B0,
$$(\theta [{\alpha _k}] + {\alpha _{\pi (k)}}\vert {\alpha _q}) = \sum\limits_{j = r + 1}^l {a_k^j} \,({\alpha _j}\vert {\alpha _q}){.}$$
(6.148)
Solving these equations for the unknown coefficients \(a_k^j\), is always possible because the Killing metric is nondegenerate on B0.

The black roots of a Tits-Satake diagram represent B0 and constitute the Dynkin diagram of the compact part \({\mathfrak m}\) of the centralizer of \({\mathfrak a}\). Because m is compact, it is the direct sum of a semi-simple compact Lie algebra and one-dimensional, Abelian \({{\mathfrak u}(1)}\) summands. The rank of \({\mathfrak m}\) (defined as the dimension of its maximal Abelian subalgebra; diagonalizability is automatic here because one is in the compact case) is equal to the sum of the rank of its semi-simple part and of the number of \({{\mathfrak u}(1)}\) terms, while the dimension of \({\mathfrak m}\) is equal to the dimension of its semi-simple part and of the number of \({{\mathfrak u}(1)}\) terms. The Dynkin diagram of \({\mathfrak m}\) reduces to the Dynkin diagram of its semi-simple part.

The rank of the compact subalgebra \({\mathfrak m}\) is given by
$${\rm{rank}}\;{\mathfrak m} = {\rm{rank}}\;{\mathfrak g} - {\rm{rank}}\;{\mathfrak p},$$
(6.149)
where rank \({\mathfrak p}\), called as we have indicated above the real rank of \({\mathfrak g}\), is given by the number of cycles of the permutation π (since two simple white roots joined by a double-arrow project on the same simple restricted root [5, 93]). These two sets of data allow one to determine the dimension of \({\mathfrak m}\) (without missing \({{\mathfrak u}(1)}\) generators) [5, 93]. Another useful information, which can be directly read off from the Tits-Satake diagrams is the dimension of the noncompact subspace \({\mathfrak p}\) appearing in the splitting \({\mathfrak g} = {\mathfrak k} \oplus {\mathfrak p}\). It is given (see Section 6.6.6) by
$${\rm{dim}}\;{\mathfrak p} = {1 \over 2}({\rm{dim}}\;{\mathfrak g} - {\rm{dim}}\;{\mathfrak m} + {\rm{rank}}\;{\mathfrak p}){.}$$
(6.150)
This can be illustrated in the two previous examples. For \({\mathfrak {su}}(3,\, 2)\), one gets dim \({\mathfrak g} = 24\), rank \({\mathfrak g} = 4\) and rank \({\mathfrak p} = 2\). It follows that rank \({\mathfrak m} = 2\) and since \({\mathfrak m}\) has no semi-simple part (no black root), it reduces to \({\mathfrak m} = {{\mathfrak u}(1)} \oplus {\mathfrak u} (1)\) and has dimension 2. This yields dim \({\mathfrak p} = 12\), and, by substraction, \({\rm{dim}}\, {\mathfrak k} = 12\) (\({\mathfrak k}\) is easily verified to be equal to \({\mathfrak {su}}(3) \oplus {\mathfrak {su}}(2) \oplus {\mathfrak u}(1)\) Similarly, for su(4,1), one gets dim \({\rm{dim}}\, {\mathfrak g} = 24\), rank \({\mathfrak g} = 4\) and rank \({\mathfrak p} = 1\). It follows that rank \({\mathfrak m} = 3\) and since the semi-simple part of \({\mathfrak m}\) is read from the black roots to be \({\mathfrak {su}}(3)\), which has rank two, one deduces \({\mathfrak m} = {\mathfrak {su}}(3) \oplus {\mathfrak u}(1)\) and \({\rm{dim}}\, {\mathfrak m} = 9\). This yields \({\rm{dim}}\, {\mathfrak p} = 8\), and, by substraction, \(\dim \,{\mathfrak k} = 16\) (\({\mathfrak k}\) is easily verified to be equal to \({\mathfrak {su}}(4) \oplus {\mathfrak u}(1)\) in this case).
Finally, from the knowledge of θ, we may obtain the restricted root space by projecting the root space according to
$$\Delta \rightarrow \bar \Delta :\alpha \mapsto \bar \alpha = {1 \over 2}(\alpha - \theta [\alpha ])$$
(6.151)
and restricting their action on \({\mathfrak a}\) since α and −θ(α) project on the same restricted root [5, 93].

6.6.5 Illustration: F4

The Lie algebra F4 is a 52-dimensional simple Lie algebra of rank 4. Its root vectors can be expressed in terms of the elements of an orthonormal basis {e k k = 1, …, 4} of a four-dimensional Euclidean space:
$${\Delta _{{F_4}}} = \left\{{\pm {e_i} \pm {e_j}\vert i < j\} \cup \{\pm {e_i}\} \cup \{{1 \over 2}(\pm {e_1} \pm {e_2} \pm {e_3} \pm {e_4})} \right\}{.}$$
(6.152)
A basis of simple roots is
$${\alpha _1} = {e_2} - {e_3},\qquad {\alpha _2} = {e_3} - {e_4},\qquad {\alpha _3} = {e_4},\qquad {\alpha _4} = {1 \over 2}({e_1} - {e_2} - {e_3} - {e_4}).$$
(6.153)
The corresponding Dynkin diagram can be obtained from Figure 37 by ignoring the painting of the vertices. To the real Lie algebra, denoted F II in [28], is associated the Tits-Satake diagram of the left hand side of Figure 37. We immediately obtain from this diagram the following information:
$${\rm{rank}}\;{\mathfrak p} = 1,\qquad {\rm{rank}} \;{\mathfrak m}= 3,\qquad {\mathfrak m}={\mathfrak s}{\mathfrak o} (7),\qquad \dim \;{\mathfrak p} = {1 \over 2}(52 - 21 + 1) = 16.$$
(6.154)
Accordingly, F II has signature \(- 21\,({\rm{compact}}\,{\rm{part}}) + ({\rm{rank}}\, {\rm{of}}\,{\mathfrak p}) = - 20\) and is denoted F4(−20). Moreover, solving a system of three equations, we obtain: θ[α4] = −α4α1 − 2 α2 − 3 α3, i.e.,
$$\theta [{e_1}] = - {e_1}\qquad {\rm{and}}\qquad \theta [{e_k}] = {e_k}\quad {\rm{if}}\,k = 2,3,4.$$
(6.155)
This shows that the projection defining the reduced root system Σ consists of projecting any given root orthogonally onto its e1 component. Thus we obtain \(\Sigma = \{\pm {1 \over 2}{e_1},\, \pm {e_1}\}\), with multiplicity 8 for \({1 \over 2}{e_1}\) (resulting from the projection of the eight roots \(\{{1 \over 2}({e_1} \pm {e_2} \pm {e_3} \pm {e_4})\}\) and 7 for e1 (resulting from the projection of the seven roots {e1 ± e k k = 2, 3, 4} ∪ {e1}).
Figure 37

On the left, the Tits-Satake diagram of the real form F4(−20). On the right, a non-admissible Tits-Satake diagram.

Let us mention that, contrary to the Vogan diagrams, any “formal Tits-Satake diagram” is not admissible. For instance if we consider the right hand side diagram of Figure 37 we get
$$\theta [{e_1}] = - {e_2},\,\theta [{e_2}] = - {e_1},\qquad {\rm{and}}\qquad \theta [{e_k}] = {e_k}\qquad {\rm{if}}\,k = 3\,{\rm{or}}\,4.$$
(6.156)
But this means that for the root α = e1, α + θ*[α] = e1e2 is again a root, which is impossible as we shall see below.

6.6.6 Some more formal considerations

Let us recall some crucial aspects of the discussion so far. Let \({{\mathfrak g}_\sigma}\) be a real form of the complex semi-simple Lie algebra \({{\mathfrak g}^{\mathbb C}}\) and σ be the conjugation it defines. We have seen that there always exists a compact real Lie algebra \({{\mathfrak u}_\tau}\) such that the corresponding conjugation τ commutes with σ. Moreover, we may choose a Cartan subalgebra \({\mathfrak h}\) of \({{\mathfrak u}_\tau}\) such that its complexification \({{\mathfrak h}^{\mathbb C}}\) is invariant under σ, i.e., \(\sigma ({{\mathfrak h}^{\mathbb C}}) = {{\mathfrak h}^{\mathbb C}}\). Then the real form \({{\mathfrak g}_\sigma}\) is said to be normally related to \(({\mathfrak u}_\theta,\, {\mathfrak h})\). As previously, we denote by the same letter θ the involution defined by duality on \(({\mathfrak h}^{\mathbb C}){\ast}\) (and also on the root lattice with respect to \({\mathfrak h}^{\mathbb C}: {\Delta}\)) by θ = τσ.

When \({{\mathfrak g}_\sigma}\) and \({{\mathfrak u}_\tau}\) are normally related, we may decompose the former into compact and non-compact components \({\mathfrak g}_{\sigma} = {\mathfrak k} \oplus {\mathfrak p}\) such that \({{\mathfrak u}_\tau} = {\mathfrak k} \oplus i{\mathfrak p}\). As mentioned, the starting point consists of choosing a maximally Abelian noncompact subalgebra \({\mathfrak a} \subset {\mathfrak p}\) and extending it to a Cartan subalgebra \({\mathfrak h} = {\mathfrak t} \oplus {\mathfrak a}\), where \({\mathfrak t} \subset {\mathfrak k}\). This Cartan subalgebra allows one to consider the real Cartan subalgebra
$${\mathfrak h^\mathbb R} = i\mathfrak t \oplus \mathfrak p = \sum\limits_{\alpha \in \Delta} {\mathbb R\,{H_\alpha}.}$$
(6.157)
Let us remind the reader that, in this case, the Cartan involution θ = στ = τσ is such that \(\theta {\vert_{\mathfrak k}} = + 1\) and \(\theta {\vert_{\mathfrak p}} = - 1\). From Equation (6.69) we obtain
$$\theta ({E_\alpha}) = {\rho _\alpha}\,{E_{\theta [\alpha ]}},$$
(6.158)
and using θ2 = 1 we deduce that
$${\rho _\alpha}\,{\rho _{\theta [\alpha ]}} = 1.$$
(6.159)
Furthermore, Equation (6.32) and the fact that the structure constants are rational yield the following relations:
$$\begin{array}{*{20}c} {{\rho _\alpha}\,{\rho _\beta}{N_{\theta [\alpha ],\,\theta [\beta ]}} = {\rho _{\alpha + \beta}}{N_{\alpha ,\,\beta}},} \\ {\quad \,\,\theta ({H_\alpha}) = {H_{\theta [\alpha ]}},} \\ {\,\,\,{\rho _\alpha}\,{\rho _{- \alpha}} = 1.\quad} \\ \end{array}$$
(6.160)
On the other hand, the commutativity of τ and σ implies
$$\sigma ({H_\alpha}) = - {H_{\sigma [\alpha ]}},\qquad \sigma ({E_\alpha}) = {\kappa _\alpha}\,{E_{\sigma [\alpha ]}},$$
(6.161)
with
$${\kappa _\alpha} = - {\bar \rho _\alpha},\qquad \sigma [\alpha ] = - \theta [\alpha ].$$
(6.162)
In particular, if the root α belongs to Δ0, defined in Equation (6.146), then θ[α] = α and thus \(\rho _\alpha ^2 = 1\), i.e.,
$${\rho _\alpha} = - {\kappa _\alpha} = \pm 1.$$
(6.163)
Let us denote by Δ0, and Δ0,+ the subsets of Δ0 corresponding to the imaginary noncompact and imaginary compact roots, respectively. We have
$${\Delta _{0,\, -}} = \{\alpha \in {\Delta _0}\vert {\rho _\alpha} = - 1\} \qquad {\rm{and}}\qquad {\Delta _{0,\, +}} = \{\alpha \in {\Delta _0}\vert {\rho _\alpha} = + 1\}$$
(6.164)
Obviously, for α ∈ Δ0, −, E α belongs to \({{\mathfrak p}^{\mathbb C}}\), while for α ∈ Δ0, +, E α belongs to \({{\mathfrak k}^{\mathbb C}}\). Moreover, if α ∈ Δ Δ0 we find
$${E_\alpha} + \theta ({E_\alpha}) \in {\mathfrak k^\mathbb C}\qquad {\rm{and}}\qquad {E_\alpha} - \theta ({E_\alpha}) \in {\rm{}}{\mathfrak p^\mathbb C}.$$
(6.165)
These remarks lead to the following explicit constructions of the complexifications of \({\mathfrak k}\) and \({\mathfrak p}\):
$$\begin{array}{*{20}c} {{\mathfrak k^\mathbb C} = {\mathfrak t^\mathbb C} \oplus \underset {\alpha \in {\Delta _{0, +}}} \oplus \,\mathbb C \,{E_\alpha} \oplus \underset {\alpha \in \Delta \backslash {\Delta _{0}}} \oplus \mathbb C \,({E_\alpha} + \theta ({E_\alpha})),} \\ {{\mathfrak p^\mathbb C} = {\mathfrak a^\mathbb C} \oplus \underset {\alpha \in {\Delta _{0, -}}} \oplus \mathbb C \,{E_\alpha} \oplus \underset {\alpha \in \Delta \backslash {\Delta _0}} \oplus \mathbb C \,({E_\alpha} - \theta ({E_\alpha})).} \\ \end{array}$$
(6.166)
Furthermore, since θ fixes all the elements of Δ0, the subspace \({\oplus _{\alpha \in {\Delta _{0,\, -}}}}\,{\mathbb C}{E_\alpha}\) belongs to the centralizer29 of \({\mathfrak a}\) and thus is empty if \({\mathfrak a}\) is maximally Abelian in \({\mathfrak p}\). Taking this remark into account, we immediately obtain the dimension formulas (6.149, 6.150).
Using, as before, the basis in Equation (6.147) we obtain for the roots belonging to B B0, i.e., for an index ir:
$$- \theta [{\alpha _i}] = \sum\limits_{j = 1, \ldots ,r} p_i^j{\alpha _j} + \sum\limits_{j = r + 1, \ldots ,l} q_i^j{\alpha _j}\qquad {\rm{with}}\qquad p_i^j,\,q_i^j \in {\rm{\mathbb N}}.$$
(6.167)
Thus
$${\alpha _i} = {(- \theta)^2})[{\alpha _i}] = \sum\limits_{\begin{array}{*{20}c} {j = 1,\, \ldots ,\,r} \\ {k = 1,\, \ldots ,\,r} \\ \end{array}} {p_i^jp_j^k{\alpha _k} +} \sum\limits_{\begin{array}{*{20}c} {j = 1,\, \ldots ,\,r} \\ {k = r + 1,\, \ldots ,\,l} \\ \end{array}} {p_i^jp_j^k{\alpha _k} -} \sum\limits_{j = r + 1,\, \ldots ,\,l} {q_i^j{\alpha _j}.}$$
(6.168)
As \(\sum\nolimits_{j = 1,\, \ldots, \, r} {p_i^jp_j^k = \delta _i^j}\), where the coefficients \({p_i^j}\) are non-negative integers, the matrix \(({p_i^j})\) must be a permutation matrix and it follows that
$$\theta [{\alpha _i}] = - {\alpha _{{\bf{\pi}}(i)}}\qquad \left({\bmod {\Delta _0}} \right),$$
(6.169)
where π is an involutive permutation of {1, …, r}.
A fundamental property of Δ is
$$\forall \alpha \in \Delta :\theta [\alpha ] + \alpha \not \in \Delta .$$
(6.170)
To show this, note that if α ∈ Δ0, it would imply that 2 α belongs to Δ, which is impossible for the root lattice of a semi-simple Lie algebra. If α ∈ Δ Δ0 and θ[α] + α ∈ Δ, then θ[α] + α ∈ Δ0. Thus we obtain using Equation (6.35) and taking into account that \({\mathfrak a}\) is maximal Abelian in \({\mathfrak p}\), that ρ α = +1, i.e.,
$$\sigma ({E_{\sigma [\alpha ] - \alpha}}) = + {E_{\theta [\alpha ] - \alpha}}$$
(6.171)
and
$$\begin{array}{*{20}c} {\left[ {{E_\alpha},\,\sigma ({E_{- \alpha}})} \right] = {\rho _{- \alpha}}\,{N_{\alpha , - \sigma [\alpha ]}}\,{E_{\alpha - \sigma [\alpha ]}},} \\ {\left[ {\sigma ({E_\alpha}),\,{E_{- \alpha}}} \right] = \overline {{\rho _{- \alpha}}} \,{N_{\alpha , - \sigma [\alpha ]}}\,{E_{\sigma [\alpha ] - \alpha}}\,} \\ {\quad \quad \quad \quad \quad \, = {\rho _{- \alpha}}\,{N_{\sigma [\alpha ], - \alpha}}\,{E_{\sigma [\alpha ] - \alpha}}.} \\ \end{array}$$
(6.172)
From this result we deduce
$${\rho _\alpha}\,{N_{\sigma [\alpha ], - \alpha}} = \overline {{\rho _{- \alpha}}} \,{N_{\alpha , - \sigma [\alpha ]}} = - {\rho _\alpha}\,{N_{\alpha , - \sigma [\alpha ]}},$$
(6.173)
i.e., \({\rho _\alpha} = - \overline {\rho - \alpha}\) which is incompatible with equation (6.160). Thus, the statement (6.170) follows.

6.7 The real semi-simple algebras \(\mathfrak{so}(k,\,l)\)

The dimensional reduction from 10 to 3 dimensions of \({\mathcal N} = 1\) supergravity coupled to m Maxwell multiplets leads to a nonlinear sigma model \({\mathcal G}/{\mathcal K}({\mathcal G})\) with \(\mathrm{Lie}(\mathcal{G})=\mathfrak{so}(8,\, 8+m)\) (see Section 7). To investigate the geometry of these cosets, we shall construct their Tits-Satake diagrams.

The \(\mathfrak{so}(n,\, \mathbb{C})\) Lie algebra can be represented by n × n antisymmetric complex matrices. The compact real form is \(\mathfrak{so}(k+l,\, \mathbb{R})\), naturally represented as the set of n × n antisymmetric real matrices. One way to describe the real subalgebras \(\mathfrak{so}(k,\, l)\), aligned with the compact form \(\mathfrak{so}(k+l,\, \mathbb{R})\), is to consider \(\mathfrak{so}(k,\, l)\) as the set of infinitesimal rotations expressed in Pauli coordinates, i.e., to represent the hyperbolic space on which they act as a Euclidean space whose first k coordinates, x a , are real while the last l coordinates y b are purely imaginary. Writing the matrices of \(\mathfrak{so}(k,\, l)\) in block form as
$$X = \left({\begin{array}{*{20}c} {\,\,A} & {i\,C} \\ {- i\,{C^t}} & B \\ \end{array}} \right),$$
(6.174)
where
$$A = - {A^t} \in {{\mathbb R}^{k \times k}},\qquad B = - {B^t} \in {{\mathbb R}^{l \times l}},\qquad C \in {{\mathbb R}^{k \times l}},$$
(6.175)
we may obtain a maximal Abelian subspace \(\mathfrak{a}\) by allowing C to have nonzero elements only on its diagonal, i.e., to be of the form:
$$C = \left({\begin{array}{*{20}c} {{a_1}} & \cdots & 0 \\ {} & \ddots & {} \\ 0 & \cdots & {{a_l}} \\ \vdots & \cdots & \vdots \\ \end{array}} \right)\quad or\quad C = \left({\begin{array}{*{20}c} {{a_1}} & \cdots & {} & \cdots & 0 \\ {} & \ddots & {} & {} & {} \\ 0 & \cdots & {{a_k}} & \cdots & 0 \\ \end{array}} \right),$$
(6.176)
with k > l or l < k, respectively.
To proceed, let us denote by H j the matrices whose entries are everywhere vanishing except for a 2 × 2 block,
$$\left({\begin{array}{*{20}c} 0 & 1 \\ {- 1} & 0 \\ \end{array}} \right),$$
, on the diagonal. These matrices have the following realisation in terms of the \({K^i}_j\) (defined in Equation (6.83)):
$${H_j} = {K^{2j - 1}}_{2j} - {K^{2j}}_{2j - 1}.$$
(6.177)
They constitute a set of \(\mathfrak{so}(k + l)\) commuting generators that provide a Cartan subalgebra; it will be the Cartan subalgebra fixed by the Cartan involution defined by the real forms that we shall now discuss.

6.7.1 Dimensions l = 2q +1 < k = 2p

Motivated by the dimensional reduction of supergravity, we shall assume k = 2p, even. We first consider l = 2q + 1 < k. Then by reordering the coordinates as follows,
$$\{{x_1},{y_1}; \cdots ;\,{x_l},{y_l};\,{x_{l + 1}},{x_{l + 2}}; \cdots ;\,{x_{2p - 2}},{x_{2p - 1}};\,{x_{2p}}\} ,$$
(6.178)
we obtain a Cartan subalgebra of \(\mathfrak{so}(2q + 1,\,2p)\), with noncompact generators first, and aligned with the one introduced in Equation (6.177) by considering the basis {i H1, ⋯, i H l , Hl+1, ⋯, Hq+p}30. These generators are all orthogonal to each other. Let us denote the elements of the dual basis by {f A A = 1, ⋯, p + q}, and split them into two subsets: {f a a = 1, ⋯, 2q + 1} and {f α α = 2q + 2, ⋯, p + q}. The action of the Cartan involution on these generators is very simple,
$$\theta [{f_a}] = - {f_a},\qquad {\rm{and}}\qquad \theta [{f_\alpha}] = + {f_\alpha}.$$
(6.179)
The root system of \(\mathfrak{so}(2q + 1,\,2p)\) is B(p+q), represented by Δ = {±f A ± f B A < B = 1, ⋯, p + q} ∪ {±f A A1, ⋯, p + q}. A simple root basis can be taken as:
$$\left\{{{\alpha _1} = {f_1} - {f_2},\, \cdots ,\,{\alpha _{p + q - 1}} = {f_{p + q - 1}} - {f_{p + q,}}\,{\alpha _{p + q}} = {f_{p + q}}} \right\}.$$
It is then straigthforward to obtain the action of the Cartan involution on the simple roots:
$$\begin{array}{*{20}c} {\theta \left[ {{\alpha _A}} \right] = - {\alpha _A}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\rm{for}}\,A = 1,\, \cdots ,\,2q,\quad \quad \quad} \\ {\theta \left[ {{\alpha _{2q + 1}}} \right] = - {\alpha _{2q + 1}} - 2\left({{\alpha _{2q + 2}} + \cdots + {\alpha _{q + p}}} \right),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\theta \left[ {{\alpha _A}} \right] = + {\alpha _A}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\rm{for}}\,A = 2q + 2,\, \cdots ,\,q + p.\quad} \\ \end{array}$$
The corresponding Tits-Satake diagrams are displayed in Figure 38.
Figure 38

Tits-Satake diagrams for the \(\mathfrak{so}(2p,\,2q + 1)\) Lie algebra with q < p. If p = q + 1, all nodes are white.

From Equation (6.179) we also obtain without effort that the set of restricted roots consists of the 4q(2q + 1) roots {±f a ± f b }, each of multiplicity one, and the 4q + 2 roots {±f a }, each of multiplicity 2(pq) − 1. These constitute a B2q+1 root system.

6.7.2 Dimensions l = 2q + 1 > k = 2p

Following the same procedure as for the previous case, we obtain a Cartan subalgebra consisting of 2p noncompact generators and qp compact generators. The corresponding Tits-Satake diagrams are displayed in Figure 39.
Figure 39

Tits-Satake diagrams for the \(\mathfrak{so}(2p,\,2q + 1)\) Lie algebra with qp. If q = p, all nodes are white.

The restricted root system is now of type B2p, with 4p(2p − 1) long roots of multiplicity one and 4p short roots of multiplicity 2(qp) + 1.

6.7.3 Dimensions l = 2q, k = 2p

Here the root system is of type Dp+q, represented by Δ = {±f A ± f B A < B = 1, ⋯, p + q}, where the orthonormal vectors f A again constitute a basis dual to the natural Cartan subalgebra of \(\mathfrak{so}(k+l)\). Now, k = 2p and l = 2q are both assumed even, and we may always suppose kl. The Cartan involution to be considered acts as previously on the f A :
$$\theta [{f_a}] = - {f_a},\qquad a = 1, \cdots ,\,2q$$
(6.180)
and
$$\theta [{f_\alpha}] = + {f_\alpha},\qquad \alpha = 2q + 1, \cdots ,\,p + q\qquad {\rm{for}}\,q < p.$$
(6.181)
The simple roots can be chosen as
$$\{{\alpha _1} = {f_1} - {f_2}, \cdots ,\,{\alpha _{p + q - 1}} = {f_{p + q - 1}} - {f_{p + q}},\,{\alpha _{p + q}} = {f_{p + q - 1}} + {f_{p + q}}\} ,$$
on which the Cartan involution has the following action:
  • For q = p
    $$\theta [{\alpha _A}] = - {\alpha _A}\qquad {\rm{for}}\,A = 1,\, \cdots ,\,q + p.$$
    (6.182)
  • For q = p − 1
    $$\begin{array}{*{20}c} {\theta [{\alpha _A}] = - {\alpha _A}\quad \quad \quad \quad {\rm{for}}\,A = 1,\, \cdots ,\,2q = q + p - 1,} \\ {\theta [{\alpha _{q + p - 1}}] = - {\alpha _{q + p}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\theta [{\alpha _{q + p}}] = - {\alpha _{q + p - 1}}.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
    (6.183)
  • For q < p − 1
    $$\begin{array}{*{20}c} {\theta [{\alpha _A}] = - {\alpha _A}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad A = 1,\, \cdots ,\,\,2q - 1,\quad \quad \quad \quad} \\ {\theta [{\alpha _{2q}}] = - {\alpha _{2q}} - 2({\alpha _{2q + 1}} + \cdots ,{\alpha _{q + p - 2}})\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {- {\alpha _{q + p - 1}} - {\alpha _{q + p}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\theta [{\alpha _A}] = + {\alpha _A}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad A = 2q + 1, \cdots ,\,q + p,\quad \quad \quad} \\ \end{array}$$
    (6.184)
The corresponding Tits-Satake diagrams are obtained in the same way as before and are displayed in Figure 40
Figure 40

Tits-Satake diagrams for the \(\mathfrak{so}(2p,\,2q)\) Lie algebra with q < p − 1, q = p − 1 and q = p.)

When q < p, the restricted root system is again of type B2q, with 4q(2q − 1) long roots of multiplicity one and 4q short roots of multiplicity 2(pq). For p = q, the short roots disappear and the restricted root system is of D2p type, with all roots having multiplicity one.

6.8 Summary — Tits-Satake diagrams for non-compact real forms

To summarize the analysis, we provide the Tits-Satake diagrams for all noncompact real forms of all simple Lie algebras [5, 93]. We do not give explicitly the Tits-Satake diagrams of the compact real forms as these are simply obtained by painting in black all the roots of the standard Dynkin diagrams.

Theorem: The simple real Lie algebras are:
  • The Lie algebras \(\mathfrak{g}^{\mathbb{R}}\) where \(\mathfrak{g}\) is one of the complex simple Lie algebras A n (n ≥ 1), B n (n ≥ 2), C n (n ≥ 3), D n (n ≥ 4), G2, F4, E6, E7, or E8, and the compact real forms of these.

  • The classical real Lie algebras of types \(\mathfrak{su},\,\mathfrak{so},\,\mathfrak{sp}\) and \(\mathfrak{sl}\). These are listed in Table 26.

  • The twelve exceptional real Lie algebras, listed in Table 27 (our conventions are due to Cartan).

Table 26

All classical real Lie algebras of \({\mathfrak s}{\mathfrak u}\), \({\mathfrak s}{\mathfrak o}\), \({\mathfrak s}{\mathfrak p}\) and \({\mathfrak s}{\mathfrak l}\) type.

Algebra

Real rank

Restricted root lattice

\({\mathfrak s}{\mathfrak u}(p,\,q)\quad p \geq q > 0p + q \geq 2\)

pq > 0 p + q ≥ 2

q

(BC) q if p > q, C q if p = q

\({\mathfrak s}{\mathfrak o}(p,\,q)\quad p > q > 0p + q = 2n + 1 \geq 5\)

p > q > 0 p + q = 2n + 1 ≥ 5

q

B q

 

pq > 0 p + q = 2n ≥ 8

q

B q if p > q, D q if p = q

\({\mathfrak s}{\mathfrak p}(p,\,q)\quad p \geq q > 0p + q \geq 3\)

pq > 0 p + q > 3

q

(BC) q if p > q, C q if p = q

\({\mathfrak s}{\mathfrak p}(n,\,{\mathbb R})\quad n \geq 3\)

n ≥ 3

n

C n

\({\mathfrak s}{\mathfrak o}\ast(2n)\quad n \geq 5\)

n ≥ 5

[n/2]

\({C_{{n \over 2}}}\) if n even, \({(BC)_{{{n - 1} \over 2}}}\) if n odd

\({\mathfrak s}{\mathfrak l}(n,\,{\mathbb R})\quad n \geq 3\)

n ≥ 3

n − 1

A n −1

\({\mathfrak s}{\mathfrak l}(n,\,{\mathbb H})\quad n \geq 2\)

n ≥ 2

n − 1

A n −1

Table 27

All exceptional real Lie algebras.

Algebra

Real rank

Restricted root lattice

G

2

G 2

F I

4

F 4

F II

1

(BC)1

E I

6

E 6

E II

4

F 4

E III

2

(BC)2

E IV

2

A 2

E V

7

E 7

E VI

4

F 4

E VII

3

C 3

E VIII

8

E 8

E IX

4

F 4

7 Kac-Moody Billiards II — The Case of Non-Split Real Forms

We will now make use of the results from the previous section to extend the analysis of Kac-Moody billiards to include also theories whose U-duality symmetries are described by algebras \(\mathfrak{u}_3\) that are non-split. The key concepts are that of restricted root systems, restricted Weyl group — and the associated concept of maximal split subalgebra — as well as the Iwasawa decomposition already encountered above. These play a prominent role in our discussion as they determine the billiard structure. We mainly follow [95].

7.1 The restricted Weyl group and the maximal split “subalgebra”

Let \(\mathfrak{u}_3\) be any real form of the complex Lie algebra \(\mathfrak{u}_3^{\mathbb{C}},\,\theta\), its Cartan involution, and let
$${{\mathfrak u}_3} = {{\mathfrak k}_3} \oplus {{\mathfrak p}_3}$$
(7.1)
be the corresponding Cartan decomposition. Furthermore, let
$${{\mathfrak h}_3} = {{\mathfrak k}_3} \oplus {{\mathfrak a}_3}$$
(7.2)
be a maximal noncompact Cartan subalgebra, with \(\mathfrak{t}_3\) (respectively, \(\mathfrak{a}_3\)) its compact (respectively, noncompact) part. The real rank of \(\mathfrak{u}_3\) is, as we have seen, the dimension of \(\mathfrak{a}_3\). Let now Δ denote the root system of \(\mathfrak{u}_{3}^{\mathbb{C}},\,\Sigma\), the restricted root system and mλ the multiplicity of the restricted root λ.

As explained in Section 4.9.2, the restricted root system of the real form \(\mathfrak{u}_{3}\) can be either reduced or non-reduced. If it is reduced, it corresponds to one of the root systems of the finite-dimensional simple Lie algebras. On the other hand, if the restricted root system is non-reduced, it is necessarily of (BC) n -type [93] (see Figure 19 for a graphical presentation of the BC3 root system).

7.1.1 The restricted Weyl group

By definition, the restricted Weyl group is the Coxeter group generated by the fundamental reflections, Equation (4.55), with respect to the simple roots of the restricted root system. The restricted Weyl group preserves multiplicities [93].

7.1.2 The maximal split “subalgebra” \(\mathfrak{f}\)

Although multiplicities are an essential ingredient for describing the full symmetry \(\mathfrak{u}_3\), they turn out to be irrelevant for the construction of the gravitational billiard. For this reason, it is useful to consider the maximal split “subalgebra” \(\mathfrak{f}\), which is defined as the real, semi-simple, split Lie algebra with the same root system as the restricted root system as \(\mathfrak{u}_3\) (in the (BC) n -case, we choose for definiteness the root system of \(\mathfrak{f}\) to be of B n -type). The real rank of \(\mathfrak{f}\) coincides with the rank of its complexification \(\mathfrak{f}^{\mathbb{C}}\), and one can find a Cartan subalgebra \(\mathfrak{h}_{\mathfrak{f}}\) of \(\mathfrak{f}\), consisting of all generators of \(\mathfrak{h}_3\) which are diagonalizable over the reals. This subalgebra \(\mathfrak{h}_{\mathfrak{f}}\) has the same dimension as the maximal noncompact subalgebra \(\mathfrak{a}_3\) of the Cartan subalgebra \(\mathfrak{h}_3\) of \(\mathfrak{u}_3\).

By construction, the root space decomposition of \(\mathfrak{f}\) with respect to \(\mathfrak{h}_{\mathfrak{f}}\) provides the same root system as the restricted root space decomposition of \(\mathfrak{u}_3\) with respect to \(\mathfrak{a}_3\), except for multiplicities, which are all trivial (i.e., equal to one) for \(\mathfrak{f}\). In the (BC) n -case, there is also the possibility that twice a root of \(\mathfrak{f}\) might be a root of \(\mathfrak{u}_3\). It is only when \(\mathfrak{u}_3\) is itself split that \(\mathfrak{f}\) and \(\mathfrak{u}_3\) coincide.

One calls \(\mathfrak{f}\) the “split symmetry algebra”. It contains as we shall see all the information about the billiard region [95]. How \(\mathfrak{f}\) can be embedded as a subalgebra of \(\mathfrak{u}_3\) is not a question that shall be of our concern here.

7.1.3 The Iwasawa decomposition and scalar coset Lagrangians

The purpose of this section is to use the Iwasawa decomposition for \(\mathfrak{u}_3\) described in Section 6.4.5 to derive the scalar Lagrangian based on the coset space \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\). The important point is to understand the origin of the similarities between the two Lagrangians in Equation (5.45) and Equation (7.8) below.

The full algebra \(\mathfrak{u}_3\) is subject to the root space decomposition
$${{\mathfrak u}_3} = {g_0} \oplus \underset {\lambda \in \Sigma} \bigoplus \,{g_\lambda}$$
(7.3)
with respect to the restricted root system. For each restricted root λ, the space gλ has dimension mλ. The nilpotent algebra \(\mathfrak{n}_{3} \subset \mathfrak{u}_3\), consisting of positive root generators only, is the direct sum
$${{\mathfrak n}_3} = \underset {\lambda \in \Sigma +} \oplus \,{g_\lambda}$$
(7.4)
over positive roots. The Iwasawa decomposition of the U-duality algebra \(\mathfrak{u}_3\) reads
$${{\mathfrak u}_3} = {{\mathfrak k}_3} \oplus {{\mathfrak a}_3} \oplus {{\mathfrak n} _3}$$
(7.5)
(see Section 6.4.5). It is \(\mathfrak{a}_3\) that appears in Equation (7.5) and not the full Cartan subalgebra \(\mathfrak{h}_3\) since the compact part of \(\mathfrak{h}_3\) belongs to \(\mathfrak{k}_3\).
This implies that when constructing a Lagrangian based on the coset space \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\), the only part of \(\mathfrak{u}_3\) that will show up in the Borel gauge is the Borel subalgebra
$${{\mathfrak b}_3} = {{\mathfrak a}_3} \oplus {{\mathfrak n}_3}.$$
(7.6)
Thus, there will be a number of dilatons equal to the dimension of \(\mathfrak{a}_3\), i.e., equal to the real rank of \(\mathfrak{u}_3\), and axion fields for the restricted roots (with multiplicities).
More specifically, an (x-dependent) element of the coset space \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\) takes the form
$${\mathcal V}(x) = {\rm{Exp}}\left[ {\phi (x)\cdot{{\mathfrak n}_3}} \right]\,{\rm{Exp}}\left[ {\chi (x)\cdot{{\mathfrak n}_3}} \right],$$
(7.7)
where the dilatons ϕ and the axions χ are coordinates on the coset space, and where x denotes an arbitrary set of parameters on which the coset element might depend. The corresponding Lagrangian becomes
$${{\mathcal L}_{{{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})}} = \sum\limits_{i = 1}^{\dim {\mathfrak a_3}} {{\partial _x}{\phi ^{(i)}}(x){\partial _x}{\phi ^{(i)}}} (x) + \sum\limits_{\alpha \in \Sigma} {\sum\limits_{{s_a} = 1}^{{\rm{mult}}\,\alpha} {{e^{2\alpha (\phi)}}} \left[ {{\partial _x}\chi _{[{s_\alpha}]}^{(\alpha)}(x) + \ldots} \right]\left[ {{\partial _x}\chi _{[{s_\alpha}]}^{(\alpha)}(x) + \ldots} \right]}$$
(7.8)
where the sums over s α = 1, ⋯, mult α are sums over the multiplicities of the positive restricted roots α.

By comparing Equation (7.8) with the corresponding expression (5.45) for the split case, it is clear why it is the maximal split subalgebra of the U-duality algebra that is relevant for the gravitational billiard. Were it not for the additional sum over multiplicities, Equation (7.8) would exactly be the Lagrangian for the coset space \({\mathcal F}/{\mathcal K}({\mathcal F})\), where \(\mathfrak{k}_{\mathfrak{f}}=\mathrm{Lie}\, \mathcal{K}(\mathcal{F})\) is the maximal compact subalgebra of \(\mathfrak{f}\) (note that \(\mathfrak{k}_{\mathfrak{f}}\neq \mathfrak{k}_{3}\)). Recall now that from the point of view of the billiard, the positive roots correspond to walls that deflect the particle motion in the Cartan subalgebra. Therefore, multiplicities of roots are irrelevant since these will only result in several walls stacked on top of each other without affecting the dynamics. (In the (BC) n -case, the wall associated with 2λ is furthermore subdominant with respect to the wall associated with λ when both λ and 2λ are restricted roots, so one can keep only the wall associated with λ. This follows from the fact that in the (BC) n -case the root system of \(\mathfrak{f}\) is taken to be of B n -type.)

7.2 “Split symmetry controls chaos”

The main point of this section is to illustrate and explain the statement “split symmetry controls chaos” [95]. To this end, we will now extend the analysis of Section 5 to non-split real forms, using the Iwasawa decomposition. As we have seen, there are two main cases to be considered:
  • The restricted root system Σ of u3 is of reduced type, in which case it is one of the standard root systems for the Lie algebras A n , B n , C n , D n , G2, F4, E6, E7 or E8.

  • The restricted root system, Σ, of \(\mathfrak{u}_3\) is of non-reduced type, in which case it is of (BC) n -type.

In the first case, the billiard is governed by the overextended algebra \(\mathfrak{f}^{++}\), where \(\mathfrak{f}\) is the “maximal split subalgebra” of \(\mathfrak{u}_3\). Indeed, the coupling to gravity of the coset Lagrangian of Equation (7.8) will introduce, besides the simple roots of \(\mathfrak{f}\) (electric walls) the affine root of \(\mathfrak{f}\) (dominant magnetic wall) and the overextended root (symmetry wall), just as in the split case (but for \(\mathfrak{f}\) instead of \(\mathfrak{u}_3\)). This is therefore a straightforward generalization of the analysis in Section 5.

The second case, however, introduces a new phenomenon, the twisted overextensions of Section 4. This is because the highest root of the (BC) n system differs from the highest root of the B n system. Hence, the dominant magnetic wall will provide a twisted affine root, to which the symmetry wall will attach itself as usual [95].

We illustrate the two possible cases in terms of explicit examples. The first one is the simplest case for which a twisted overextension appears, namely the case of pure four-dimensional gravity coupled to a Maxwell field. This is the bosonic sector of \({\mathcal N} = 2\) supergravity in four dimensions, which has the non-split real form \(\mathfrak{su}(2,\,1)\) as its U-duality symmetry. The restricted root system of \(\mathfrak{su}(2,\,1)\) is the non-reduced (BC)1-system, and, consequently, as we shall see explicitly, the billiard is governed by the twisted overextension \(A_2^{(2) +}\).

The second example is that of heterotic supergravity, which exhibits an SO(8, 24)/(SO(8) × SO(24)) coset symmetry in three dimensions. The U-duality algebra is thus \(\mathfrak{so}(8,\,24)\), which is non-split. In this example, however, the restricted root system is B8, which is reduced, and so the billiard is governed by a standard overextension of the maximal split subalgebra \(\mathfrak{so}(8,\,9)\subset \mathfrak{so}(8,\,24)\).

7.2.1 (BC)1 and \({\mathcal N} = 2,\,D = 4\) pure supergravity

We consider \({\mathcal N} = 2\) supergravity in four dimensions where the bosonic sector consists of gravity coupled to a Maxwell field. It is illuminating to compare the construction of the billiard in the two limiting dimensions, D = 4 and D = 3.

In maximal dimension the metric contains three scale factors, β1, β2 and β3, which give rise to three symmetry wall forms,
$${s_{21}}(\beta) = {\beta ^2} - {\beta ^1},\qquad {s_{32}}(\beta) = {\beta ^3} - {\beta ^2},\qquad {s_{31}}(\beta) = {\beta ^3} - {\beta ^1},$$
(7.9)
where only s21 and s32 are dominant. In four dimensions the curvature walls read
$${c_{123}}(\beta) \equiv {c_1}(\beta) = 2{\beta ^1},\qquad {c_{231}}(\beta) \equiv {c_2}(\beta) = 2{\beta ^2},\qquad {c_{312}}(\beta) \equiv {c_3}(\beta) = 2{\beta ^3}.$$
(7.10)
Finally we have the electric and magnetic wall forms of the Maxwell field. These are equal because there is no dilaton. Hence, the wall forms are
$${e_1}(\beta) = {m_1}(\beta) = {\beta ^1},\qquad {e_2}(\beta) = {m_2}(\beta) = {\beta ^2},\qquad {e_3}(\beta) = {m_3}(\beta) = {\beta ^3}.$$
(7.11)
The billiard region \({{\mathcal B}_{{{\mathcal M}_\beta}}}\) is defined by the set of dominant wall forms,
$${{\mathcal B}_{{{\mathcal M}_\beta}}} = \{\beta \in {{\mathcal M}_\beta}\,\vert \,{e_1}(\beta),{s_{21}}(\beta),{s_{32}}(\beta)\, > \,0\} {.}$$
(7.12)
The first dominant wall form, e1 (β), is twice degenerate because it occurs once as an electric wall form and once as a magnetic wall form. Because of the existence of the curvature wall, c1(β) = 2β1, we see that 2α1 is also a root.
The same billiard emerges after reduction to three spacetime dimensions, where the algebraic structure is easier to exhibit. As before, we perform the reduction along the first spatial direction. The associated scale factor is then replaced by the Kaluza-Klein dilaton \(\hat \varphi\) such that
$${\beta ^1} = {1 \over {\sqrt 2}}\hat \varphi{.}$$
(7.13)
The remaining scale factors change accordingly,
$${\beta ^2} = {\hat \beta ^2} - {1 \over {\sqrt 2}}\hat \varphi ,\qquad {\beta ^3} = {\hat \beta ^3} - {1 \over {\sqrt 2}}\hat \varphi ,$$
(7.14)
and the two symmetry walls become
$${s_{21}}(\hat \beta ,\hat \varphi) = {\hat \beta ^2} - \sqrt 2 \hat \varphi ,\qquad {\hat s_{32}}(\hat \beta) = {\hat \beta ^3} - {\hat \beta ^2}.$$
(7.15)
In addition to the dilaton \(\hat \varphi\), there are three axions: one \((\hat \chi)\) arising from the dualization of the Kaluza-Klein vector, one \(({\hat \chi ^E})\) coming from the component A1 of the Maxwell vector potential and one \(({\hat \chi ^C})\) coming from dualization of the Maxwell vector potential in 3 dimensions (see, e.g., [35] for a review). There are then a total of four scalars. These parametrize the coset space SU(2, 1)/S(U(2) × U(1)) [113].
The Einstein-Maxwell Lagrangian in four dimensions yields indeed in three dimensions the Einstein-scalar Lagrangian, where the Lagrangian for the scalar fields is given by
$${{\mathcal L}_{SU(2,1)/S(U(2) \times U(1))}} = {\partial _\mu}\hat \varphi {\partial ^\mu}\hat \varphi + {e^{2{e_1}(\hat \varphi)}}\left({{\partial _\mu}{{\hat \chi}^E}{\partial ^\mu}{{\hat \chi}^E} + {\partial _\mu}{{\hat \chi}^C}{\partial ^\mu}{{\hat \chi}^C}} \right) + {e^{4{e_1}(\hat \varphi)}}\left({{\partial _\mu}\hat \chi {\partial ^\mu}\hat \chi} \right) + \cdots$$
(7.16)
with
$${e_1}(\hat \varphi) = {1 \over {\sqrt 2}}\hat \varphi .$$
Here, the ellipses denotes terms that are not relevant for understanding the billiard structure. The U-duality algebra of \({\mathcal N} = 2\) supergravity compactified to three dimensions is therefore
$${{\mathfrak u}_3} = {\mathfrak s}{\mathfrak u}(2,1),$$
(7.17)
which is a non-split real form of the complex Lie algebra \(\mathfrak{sl}(3,\,\mathbb{C})\). This is in agreement with Table 1 of [113]. The restricted root system of \(\mathfrak{su}(2,\,1)\) is of (BC)1-type (see Table 28 in Section 6.8) and has four roots: α1, 2α1, −α1 and −2α1. One may take α1 to be the simple root, in which case Σ+ = {α1, 2α1} and 2α1 is the highest root. The short root α1 is degenerate twice while the long root 2α1 is nondegenerate. The Lagrangian (7.16) coincides with the Lagrangian (7.8) for \(\mathfrak{su}(2,\,1)\) with the identification
$${\hat \alpha _1} \equiv {e_1}.$$
(7.18)
We clearly see from the Lagrangian that the simple root \({\hat \alpha _1}\) has multiplicity 2 in the restricted root system, since the corresponding wall appears twice. The maximal split subalgebra may be taken to be \(A_{1}\equiv\mathfrak{su}(1,\,1)\) with root system \(\{{\hat \alpha _1},\, - {\hat \alpha _1}\}\).
Table 28

Tits-Satake diagrams (A n series)

A n series n ≥ 1

Tits-Satake diagram

Restricted root system

\({{\mathfrak s}{\mathfrak l}(n,\,{\mathbb R}),\,n \geq 3}\)

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A I

 

A n

\({{\mathfrak s}{\mathfrak u}\ast(n + 1),\,n = 2k + 1}\)

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A II

A 2k

\({{\mathfrak s}{\mathfrak u}(p,\,n + 1 - p)}\)

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A III

BC p

\({{\mathfrak s}{\mathfrak u}\left({{{n + 1} \over 2},\,{{n + 1} \over 2}} \right),\,n = 2k + 1}\)

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A III

 

C (k+1)

\({\mathfrak s}{\mathfrak u}(1,\,n - 1),\,n \geq 3\)

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A IV

A 1

Table 29

Tits-Satake diagrams (B n series)

B n series n ≥ 4

Tits-Satake diagram

Restricted root system

\({{\mathfrak s}{\mathfrak o}(p,\,2n - p + 1),\,p \geq 1}\)

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B I

B p

\({{\mathfrak s}{\mathfrak o}(1,\,2n)}\)

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B II

A 1

Table 30

Tits-Satake diagrams (C n series)

C n series n ≥ 3

Tits-Satake diagram

Restricted root system

\({{\mathfrak s}{\mathfrak p}(n,\,{\mathbb R})}\)

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C I

 

C n

\({{\mathfrak s}{\mathfrak p}(p,\,n - p)}\)

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C II

B p

\({{\mathfrak s}{\mathfrak p}\left({{n \over 2},\,{n \over 2}} \right),\,n = 2k}\)

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C II

 

\({C_{{n \over 2}}}\)

Table 31

Tits-Satake diagrams (D n series)

D n series n ≥ 4

Tits-Satake diagram

Restricted root system

\({{\mathfrak s}{\mathfrak o}(p,\,2n - p),\,p \leq n - 2}\)

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D I

B p

\({{\mathfrak s}{\mathfrak o}(n - 1,\,n + 1)}\)

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D I

 

B (n−1)

\({{\mathfrak s}{\mathfrak o}(n,\,n)}\)

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D I

 

D n

\({{\mathfrak s}{\mathfrak o}(1,\,2n - 1)}\)

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D II

 

A 1

\({{\mathfrak s}{\mathfrak o}(\ast(2n)),\,n = 2k}\)

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D III

 

C 2k−1

\({{\mathfrak s}{\mathfrak o}(\ast(2n)),\,n = 2k + 1}\)

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D III

 

BC 2k

Table 32

Tits-Satake diagrams (G2 series)

G2 series

Tits-Satake diagram

Restricted root system

G 2(2)

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G

Table 33

Tits-Satake diagrams (F4 series)

F4 series

Tits-Satake diagram

Restricted root system

F 4(4)

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F I

  

F 4(− 20)

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F II

  
Table 34

Tits-Satake diagrams (E6 series)

Table 35

Tits-Satake diagrams (E7 series)

E7 series

Tits-Satake diagram

Restricted root system

E 7(7)

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E V

E 7(− 5)

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E VI

E 7(− 25)

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E VII

Table 36

Tits-Satake diagrams (E8 series)

E8 series

Tits-Satake diagram

Restricted root system

E 8(8)

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E VIII

E 8(− 24)

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E IX

Let us now see how one goes from \(\mathfrak{su}(2,\,1)\) described by the scalar Lagrangian to the full algebra, by including the gravitational scale factors. Let us examine in particular how the twist arises. For the standard root system of A1 the highest root is just \({\hat \alpha _1}\). However, as we have seen, for the (BC)1 root system the highest root is \({\theta _{{{(BC)}_1}}} = 2{\hat \alpha _1}\), with
$$({\theta _{(BC)1}}\vert {\theta _{{{(BC)}_1}}}) = 4({\hat \alpha _1}\vert {\hat \alpha _1}) = 2.$$
(7.19)
So we see that because of \(({\hat \alpha _1}\vert{\hat \alpha _1}) = {1 \over 2}\), the highest root \({\theta _{{{(BC)}_1}}}\) already comes with the desired normalization. The affine root is therefore
$${{\hat \alpha}_2}(\hat \varphi ,\hat \beta) = \hat m_2^{\hat \chi}(\hat \beta ,\hat \varphi) = {{\hat \beta}^2} - {\theta _{{{(BC)}_1}}} = {{\hat \beta}^2} - \sqrt 2 \hat \varphi ,$$
(7.20)
whose norm is
$$({\hat \alpha _2}\vert {\hat \alpha _2}) = 2.$$
(7.21)
The scalar product between \({\hat \alpha _1}\) and \({\hat \alpha _2}\) is \(({\hat \alpha _1}\vert{\hat \alpha _2}) = - 1\) and the Cartan matrix at this stage becomes (i, j = 1, 2)
$${A_{ij}}[A_2^{(2)}] = 2{{({{\hat \alpha}_i}\vert {{\hat \alpha}_j})} \over {({{\hat \alpha}_i}\vert {{\hat \alpha}_i})}} = \left({\begin{array}{*{20}c} 2 & {- 4} \\ {- 1} & 2 \\ \end{array}} \right),$$
(7.22)
which may be identified not with the affine extension of A1 but with the Cartan matrix of the twisted affine Kac-Moody algebra \(A_2^{(2)}\). It is the underlying (BC)1 root system that is solely responsible for the appearance of the twist. Because of the fact that \({\theta _{{{(BC)}_1}}} = 2{\hat \alpha _1}\) the two simple roots of the affine extension come with different length and hence the asymmetric Cartan matrix in Equation (7.22). It remains to include the overextended root
$${\hat \alpha _3}(\hat \beta) = {\hat s_{32}}(\hat \beta) = {\hat s_{32}}(\hat \beta) = {\hat \beta ^3} - {\hat \beta ^2},$$
(7.23)
which has non-vanishing scalar product only with \({\hat \alpha _2},\,({\hat \alpha _2}\vert{\hat \alpha _3}) = - 1\), and so its node in the Dynkin diagram is attached to the second node by a single link. The complete Cartan matrix is
$$A[A_2^{(2) +}] = \left({\begin{array}{*{20}c} 2 & {- 4} & 0 \\ {- 1} & 2 & {- 1} \\ 0 & {- 1} & 2 \\ \end{array}} \right),$$
(7.24)
which is the Cartan matrix of the Lorentzian extension \(A_2^{(2) +}\) of \(A_2^{(2)}\) henceforth referred to as the twisted overextension of A1. Its Dynkin diagram is displayed in Figure 41.
Figure 41

The Dynkin diagram of \(A_2^{(2) +}\). Label 1 denotes the simple root \({\hat \alpha _{(1)}}\) of the restricted root system of \(\mathfrak{u}_{3}=\mathfrak{su}(2,\,1)\). Labels 2 and 3 correspond to the affine and overextended roots, respectively. The arrow points towards the short root which is normalized such that \(({\hat \alpha _1}\vert{\hat \alpha _1}) = {1 \over 2}\).

The algebra \(A_2^{(2) +}\) was already analyzed in Section 4, where it was shown that its Weyl group coincides with the Weyl group of the algebra \(A_1^{+ +}\). Thus, in the BKL-limit the dynamics of the coupled Einstein-Maxwell system in four-dimensions is equivalent to that of pure four-dimensional gravity, although the set of dominant walls are different. Both theories are chaotic.

7.2.2 Heterotic supergravity and \({\mathfrak {sl}(8,\,24)}\)

Pure \({\mathcal N} = 1\) supergravity in D = 10 dimensions has a billiard description in terms of the hyperbolic Kac-Moody algebra \(D{E_{10}} = D_8^{+ +}\) [45]. This algebra is the overextension of the U-duality algebra, \({{\mathfrak u}_3} = {D_8} = {\mathfrak {so}}(8,\,8)\), appearing upon compactification to three dimensions. In this case, \({\mathfrak {so}}(8,\,8)\) is the split form of the complex Lie algebra D8, so we have \({\mathfrak f} = {{\mathfrak u}_3}\).

By adding one Maxwell field to the theory we modify the billiard to the hyperbolic Kac-Moody algebra \(B{E_{10}} = B_8^{+ +}\), which is the overextension of the split form \({\mathfrak {so}}(8,\,9)\) of B8 [45]. This is the case relevant for (the bosonic sector of) Type I supergravity in ten dimensions. In both these cases the relevant Kac-Moody algebra is the overextension of a split real form and so falls under the classification given in Section 5.

Let us now consider an interesting example for which the relevant U-duality algebra is non-split. For the heterotic string, the bosonic field content of the corresponding supergravity is given by pure gravity coupled to a dilaton, a 2-form and an E8 × E8 Yang-Mills gauge field. Assuming the gauge field to be in the Cartan subalgebra, this amounts to adding 16 \({\mathcal N} = 1\) vector multiplets in the bosonic sector, i.e., to adding 16 Maxwell fields to the ten-dimensional theory discussed above. Geometrically, these 16 Maxwell fields correspond to the Kaluza-Klein vectors arising from the compactification on T16 of the 26-dimensional bosonic left-moving sector of the heterotic string [89].

Consequently, the relevant U-duality algebra is \({\mathfrak {so}}(8,\,8+16)={\mathfrak {so}}(8,\,24)\) which is a non-split real form. But we know that the billiard for the heterotic string is governed by the same Kac-Moody algebra as for the Type I case mentioned above, namely \(B{E_{10}} \equiv {\mathfrak {so}}{(8,\,9)^{+ +}}\), and not \({\mathfrak {so}}(8,\,24)^{+ +}\) as one might have expected [45]. The only difference is that the walls associated with the one-forms are degenerate 16 times. We now want to understand this apparent discrepancy using the machinery of non-split real forms exhibited in previous sections. The same discussion applies to the SO(32)-superstring.

In three dimensions the heterotic supergravity Lagrangian is given by a pure three-dimensional Einstein-Hilbert term coupled to a nonlinear sigma model for the coset SO(8, 24)/(SO(8) × SO(24)). This Lagrangian can be understood by analyzing the Iwasawa decomposition of \({\mathfrak {so}}(8,\,24) = {\rm{Lie}}[SO(8,\,24)]\). The maximal compact subalgebra is
$${{\mathfrak k}_3} = {\mathfrak s}{\mathfrak o}(8) \oplus {\mathfrak s}{\mathfrak o}(24).$$
(7.25)
This subalgebra does not appear in the sigma model since it is divided out in the coset construction (see Equation (7.7)) and hence we only need to investigate the Borel subalgebra \({\mathfrak {a}_3} \oplus {\mathfrak {n}_3}\) of \({\mathfrak {so}}(8,\,24)\) in more detail.

As was emphasized in Section 7.1, an important feature of the Iwasawa decomposition is that the full Cartan subalgebra \({{\mathfrak h}_3}\) does not appear explicitly but only the maximal noncompact Cartan subalgebra \({{\mathfrak a}_3}\), associated with the restricted root system. This is the maximal Abelian subalgebra of \({\mathfrak {u}_3} = {\mathfrak {so}}(8,\,24)\), whose adjoint action can be diagonalized over the reals. The remaining Cartan generators of \({{\mathfrak h}_3}\) are compact and so their adjoint actions have imaginary eigenvalues. The general case of \({\mathfrak {so}}(2q,\,2p)\) was analyzed in detail in Section 6.7 where it was found that if q < p, the restricted root system is of type B2q. For the case at hand we have q = 4 and p = 12 which implies that the restricted root system of \({\mathfrak {so}}(8,\,24)\) is (modulo multiplicities) \({\Sigma _{{\mathfrak {so}}(8,\,24)}} = {B_8}\).

The root system of B8 is eight-dimensional and hence there are eight Cartan generators that may be simultaneously diagonalized over the real numbers. Therefore the real rank of \({\mathfrak {so}}(8,\,24)\) is eight, i.e.,
$${\rm{ran}}{{\rm{{\mathbb k}}}_{\rm{R}}}\,{{\mathfrak u}_3} = \dim {{\mathfrak a}_3} = 8.$$
(7.26)
Moreover, it was shown in Section 6.7 that the restricted root system of \({\mathfrak {so}}(2p,\,2q)\) has 4q(2q − 1) long roots which are nondegenerate, i.e., with multiplicity one, and 4q long roots with multiplicities 2(pq). In the example under consideration this corresponds to seven nondegenerate simple roots α1, ⋯, α7 and one short simple root α8 with multiplicity 16. The Dynkin diagram for the restricted root system \({\Sigma _{{\mathfrak {so}}(8,\,24)}}\) is displayed in Figure 42 with the multiplicity indicated in brackets over the short root. It is important to note that the restricted root system \({\mathfrak {so}}(8,\,24)\) differs from the standard root system of \({\mathfrak {so}}(8,\,9)\) precisely because of the multiplicity 16 of the simple root α8.
Figure 42

The Dynkin diagram representing the restricted root system \({\Sigma _{{\mathfrak {so}}(8,\,24)}}\) of \({\mathfrak {so}}(8,\,24)\). Labels 1, ⋯, 7 denote the long simple roots that are nondegenerate while the eighth simple root is short and has multiplicity 16.

Because of these properties of \({\mathfrak {so}}(8,\,24)\) the Lagrangian for the coset
$${{SO(8,24)} \over {SO(8) \times SO(24)}}$$
(7.27)
takes a form very similar to the Lagrangian for the coset
$${{SO(8,9)} \over {SO(8) \times SO(9)}}.$$
(7.28)
The algebra constructed from the restricted root system B8 is the maximal split subalgebra
$${\mathfrak f} = {\mathfrak s}{\mathfrak o}(8,9){.}$$
(7.29)
Let us now take a closer look at the Lagrangian in three spacetime dimensions We parametrize an element of the coset by
$${\mathcal V}({x^\mu}) = {\rm{Exp}}\left[ {\sum\limits_{i = 1}^8 {{\phi ^{(i)}}} ({x^\mu})\alpha _i^ \vee} \right]{\rm{Exp}}\left[ {\sum\limits_{\gamma \in {\Delta _ +}} {{\chi ^{(\gamma)}}} ({x^\mu}){E_\gamma}} \right] \in {{SO(8,24)} \over {SO(8) \times SO(24)}},$$
(7.30)
where x μ (μ = 0, 1, 2) are the coordinates of the external three-dimensional spacetime, \(\alpha _i^ \vee\) are the noncompact Cartan generators and Δ+ denotes the full set of positive roots of \({\mathfrak {so}}(8,\,24)\).
The Lagrangian constructed from the coset representative in Equation (7.30) becomes (again, neglecting corrections to the single derivative terms of the form “ x χ
$$\begin{array}{*{20}c} {{{\mathcal L}_{{{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})}} = \sum\limits_{i = 1}^8 {{\partial _\mu}} {\phi ^{(i)}}(x)\,{\partial ^\mu}{\phi ^{(i)}}(x) + \sum\limits_{j = 1}^7 {{e^{{\alpha _j}(\phi)}}} \,{\partial _\mu}{\chi ^{(j)}}(x)\,{\partial ^\mu}{\chi ^{(j)}}(x)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {e^{{\alpha _8}(\phi)}}\left({\sum\limits_{k = 1}^{16} {{\partial _\mu}} \chi _{[k]}^{(8)}(x)\,{\partial ^\mu}\chi _{[k]}^{(8)}(x)} \right) + \sum\limits_{\alpha \in {{\tilde \Sigma}^ +}} \sum\limits_{{s_\alpha} = 1}^{{\rm{mult}}(\alpha)} {e^{\alpha (\phi)}}\,{\partial _\mu}\chi _{[{s_\alpha}]}^{(\alpha)}(x)\,{\partial ^\mu}\chi _{[{s_\alpha}]}^{(\alpha)}(x),\qquad} \\ \end{array}$$
(7.31)
where \({{\tilde \Sigma}^ +}\) denotes all non-simple positive roots of Σ, i.e.,
$${\tilde \Sigma ^ +} = {\Sigma ^ +}/\bar B$$
(7.32)
with
$$\bar B = \{{\alpha _1} \cdots ,{\alpha _8}\} .$$
(7.33)
This Lagrangian is equivalent to the Lagrangian for SO(8, 9)/(SO(8) × SO(9)) except for the existence of the non-trivial root multiplicities.
The billiard for this theory can now be computed with the same methods that were treated in detail in Section 5.3.3. In the BKL-limit, the simple roots α1, ⋯, α8 become the non-gravitational dominant wall forms. In addition to this we get one magnetic and one gravitational dominant wall form:
$$\begin{array}{*{20}c} {{\alpha _0} = {\beta ^1} - \theta (\phi),} \\ {{\alpha _{- 1}} = {\beta ^2} - {\beta ^1},\quad} \\ \end{array}$$
(7.34)
where θ(ϕ) is the highest root of \({\mathfrak {so}}(8,\,9)\):
$$\theta = {\alpha _1} + 2{\alpha _2} + \cdots + 2{\alpha _7} + {\alpha _8}.$$
(7.35)
The affine root α0 attaches with a single link to the second simple root α2 in the Dynkin diagram of B8. Similarly the overextended root α−1 attaches to α0 with a single link so that the resulting Dynkin diagram corresponds to BE10 (see Figure 43). It is important to note that the underlying root system is still an overextension of the restricted root system and hence the multiplicity of the simple short root α8 remains equal to 16. Of course, this does not affect the dynamics in the BKL-limit because the multiplicity of α8 simply translates to having multiple electric walls on top of each other and this does not alter the billiard motion.
Figure 43

The Dynkin diagram representing the overextension \(B_8^{+ +}\) of the restricted root system Σ = B8 of \({\mathfrak {so}}(8,\,24)\). Labels −1, 0, 1, ⋯, 7 denote the long simple roots that are nondegenerate while the eighth simple root is short and has multiplicity 16.

This analysis again showed explicitly how it is always the split symmetry that controls the chaotic behavior in the BKL-limit. It is important to point out that when going beyond the strict BKL-limit, one expects more and more roots of the algebra to play a role. Then it is no longer sufficient to study only the maximal split subalgebra \({\mathfrak {so}}{(8,\,9)^{+ +}}\) but instead the symmetry of the theory is believed to contain the full algebra \({\mathfrak {so}}{(8,\,24)^{+ +}}\). In the spirit of [47] one may then conjecture that the dynamics of the heterotic supergravity should be equivalent to a null geodesic on the coset space \(SO{(8,\,24)^{+ +}}/{\mathcal K}(SO{(8,\,24)^{+ +}})\)) [42].

7.3 Models associated with non-split real forms

In this section we provide a list of all theories coupled to gravity which, upon compactification to three dimensions, display U-duality algebras that are not maximal split [95]. This therefore completes the classification of Section 5.

One can classify the various theories through the number \({\mathcal N}\) of supersymmetries that they possess in D = 4 spacetime dimensions. All p-forms can be dualized to scalars or to 1-forms in four dimensions so the theories all take the form of pure supergravities coupled to collections of Maxwell multiplets. The analysis performed for the split forms in Section 5.3 were all concerned with the cases of \({\mathcal N} = 0\) or \({\mathcal N} = 8\) supergravity in D = 4. We consider all pure four-dimensional supergravities \(({\mathcal N} = 1,\, \cdots, \,8)\) as well as pure \({\mathcal N} = 4\) supergravity coupled to k Maxwell multiplets.

As we have pointed out, the main new feature in the non-split cases is the possible appearance of so-called twisted overextensions. These arise when the restricted root system of \({\mathcal U}_3\) is of non-reduced type hence yielding a twisted affine Kac-Moody algebra in the affine extension of \({\mathfrak f} \subset {{\mathfrak u}_3}\). It turns out that the only cases for which the restricted root system is of non-reduced ((BC)-type) is for the pure \({\mathcal N} = 2,\,3\) and \({\mathcal N} = 5\) supergravities. The example of \({\mathcal N} = 2\) was already discussed in detail before, where it was found that the U-duality algebra is \({{\mathfrak u}_3} = {\mathfrak su}(2,\,1)\) whose restricted root system is (BC)1, thus giving rise to the twisted overextension \(A_2^{(2) +}\). It turns out that for the \({\mathcal N} = 3\) case the same twisted overextension appears. This is due to the fact that the U-duality algebra is \({{\mathfrak u}_3} = {\mathfrak su}(4,\,1)\) which has the same restricted root system as \({\mathfrak {so}}(2,\,1)\), namely (BC)1. Hence, \(A_1^{(2) +}\) controls the BKL-limit also for this theory.

The case of \({\mathcal N} = 5\) follows along similar lines. In D = 3 the non-split form E6(−14) of E6 appears, whose maximal split subalgebra is \({\mathfrak f} = {C_2}\). However, the relevant Kac-Moody algebra is not \(C_2^{+ +}\) but rather \(A_4^{(2) +}\) because the restricted root system of E6(−14) is (BC)2.

In Table 37 we display the algebraic structure for all pure supergravities in four dimensions as well as for \({\mathcal N} = 0\) supergravity with k Maxwell multiplets. We give the relevant U-duality algebras \({{\mathfrak u}_3}\), the restricted root systems Σ, the maximal split subalgebras \({\mathfrak f}\) and, finally, the resulting overextended Kac-Moody algebras \({\mathfrak g}\).
Table 37

Classification of theories whose U-duality symmetry in three dimensions is described by a non-split real form \({\mathfrak u_3}\). The leftmost column indicates the number \({\mathcal N}\) of supersymmetries that the theories possess when compactified to four dimensions, and the associated number k of Maxwell multiplets. The middle column gives the restricted root system Σ of \({\mathfrak u_3}\) and to the right of this we give the maximal split subalgebras \({\mathfrak f} \subset {\mathfrak u_3}\), constructed from a basis of Σ. Finally, the rightmost column displays the overextended Kac-Moody algebras that governs the billiard dynamics.

\(({\mathcal N},\,k)\)

\({\mathfrak u_3}\)

Σ

\({\mathfrak f}\)

\({\mathfrak g}\)

(1, 0)

\({\mathfrak s}{\mathfrak l}(2,\,{\mathbb R})\)

A 1

A 1

\(A_1^{+ +}\)

(2, 0)

\({\mathfrak s}{\mathfrak u}(2,\,1)\)

(BC)1

A 1

\(A_2^{(2) +}\)

(3, 0)

\({\mathfrak s}{\mathfrak u}(4,\,1)\)

(BC)1

A 1

\(A_1^{(2) +}\)

(4, 0)

\({\mathfrak s}{\mathfrak o}(8,\,2)\)

C 2

C 2

\(C_2^{+ +}\)

(4, k < 6)

\({\mathfrak s}{\mathfrak o}(8,\,k + 2)\)

B k+2

B k+2

\(B_{k + 2}^{+ +}\)

(4, 6)

\({\mathfrak s}{\mathfrak o}(8,\,8)\)

D 8

D 8

\(D{E_{10}} = D_8^{+ +}\)

(4, k > 6)

\({\mathfrak s}{\mathfrak o}(8,\,k + 2)\)

B 8

B 8

\(B{E_{10}} = B_8^{+ +}\)

(5, 0)

E 6(−14)

(BC)2

C 2

\(A_4^{(2) +}\)

(6, 0)

E 7(−5)

F 4

F 4

\(F_4^{+ +}\)

(8, 0)

E 8(+8)

E 8

E 8

\({E_{10}} = E_8^{+ +}\)

Let us end this section by noting that the study of real forms of hyperbolic Kac-Moody algebras has been pursued in [17].

8 Level Decomposition in Terms of Finite Regular Subalgebras

We have shown in the previous sections that Weyl groups of Lorentzian Kac-Moody algebras naturally emerge when analysing gravity in the extreme BKL regime. This has led to the conjecture that the corresponding Kac-Moody algebra is in fact a symmetry of the theory (most probably enlarged with new fields) [46]. The idea is that the BKL analysis is only the “revelator” of that huge symmetry, which would exist independently of that limit, without making the BKL truncations. Thus, if this conjecture is true, there should be a way to rewrite the gravity Lagrangians in such a way that the Kac-Moody symmetry is manifest. This conjecture itself was made previously (in this form or in similar ones) by other authors on the basis of different considerations [113, 139, 167]. To explore this conjecture, it is desirable to have a concrete method of dealing with the infinite-dimensional structure of a Lorentzian Kac-Moody algebra \({\mathfrak g}\). In this section we present such a method.

The method by which we shall deal with the infinite-dimensional structure of a Lorentzian Kac-Moody algebra \({\mathfrak g}\) is based on a certain gradation of \({\mathfrak g}\) into finite-dimensional subspaces \({{\mathfrak g}_\ell}\). More precisely, we will define a so-called level decomposition of the adjoint representation of \({\mathfrak g}\) such that each level corresponds to a finite number of representations of a finite regular subalgebra \({\mathfrak r}\) of \({\mathfrak g}\). Generically the decomposition will take the form of the adjoint representation of \({\mathfrak r}\) plus a (possibly infinite) number of additional representations of \({\mathfrak r}\). This type of expansion of \({\mathfrak g}\) will prove to be very useful when considering sigma models invariant under \({\mathfrak g}\) for which we may use the level expansion to consistently truncate the theory to any finite level (see Section 9).

We begin by illustrating these ideas for the finite-dimensional Lie algebra \({\mathfrak {sl}(3,\,\mathbb R)}\) after which we generalize the procedure to the indefinite case in Sections 8.2, 8.3 and 8.4.

8.1 A finite-dimensional example: \({\mathfrak {sl}(3,\,\mathbb R)}\)

The rank 2 Lie algebra \({\mathfrak g} = {\mathfrak {sl}(3,\,{\mathbb R})}\) is characterized by the Cartan matrix
$$A[\mathfrak{sl}(3,\mathbb{R})] = \left({\begin{array}{*{20}c} 2 & {- 1}\\ {- 1} & 2 \end{array}} \right),$$
(8.1)
whose Dynkin diagram is displayed in Figure 44.
Figure 44

The Dynkin diagram of \({\mathfrak {sl}(3,\,\mathbb R)}\).

Recall from Section 6 that \({\mathfrak {sl}(3,\,\mathbb R)}\) is the split real form of \(\mathfrak {sl}(3,\,{\mathbb C}) \equiv {A_2}\), and is thus defined through the same Chevalley-Serre presentation as for \({\mathfrak {sl}(3,\,\mathbb C)}\), but with all coefficients restricted to the real numbers.

The Cartan generators {h1, h2} will indifferently be denoted by \(\{\alpha _i^ \vee, \,\alpha _i^ \vee \}\). As we have seen, they form a basis of the Cartan subalgebra \({\mathfrak h}\), while the simple roots {α1, α2}, associated with the raising operators e1 and e2, form a basis of the dual root space \({{\mathfrak h}^{\star}}\). Any root \(\gamma \in {{\mathfrak h}^{\star}}\) can thus be decomposed in terms of the simple roots as follows,
$$\gamma = m{\alpha _1} + \ell {\alpha _2},$$
(8.2)
and the only values of (m, n) are (1, 0), (0, 1), (1, 1) for the positive roots and minus these values for the negative ones.
The algebra \({\mathfrak {sl}(3,\,\mathbb R)}\) defines through the adjoint action a representation of \({\mathfrak {sl}(3,\,\mathbb R)}\) itself, called the adjoint representation, which is eight-dimensional and denoted 8. The weights of the adjoint representation are the roots, plus the weight (0, 0) which is doubly degenerate. The lowest weight of the adjoint representation is
$${\Lambda _\mathfrak{g}} = - {\alpha _1} - {\alpha _2},$$
(8.3)
corresponding to the generator [f1, f2]. We display the weights of the adjoint representation in Figure 45.
Figure 45

Level decomposition of the adjoint representation \({\mathcal R_{ad}} = 8\) of \({\mathfrak {sl}(3,\,\mathbb R)}\) into representations of the subalgebra \({\mathfrak {sl}(2,\,\mathbb R)}\). The labels 1 and 2 indicate the simple roots α1 and α2. Level zero corresponds to the horizontal axis where we find the adjoint representation \(\mathcal R_{\rm {ad}}^{(0)} = {3_0}\) of \({\mathfrak {sl}(2,\,\mathbb R)}\) (red nodes) and the singlet representation \({\mathcal R_s}^{(0)} = {1_0}\) (green circle about the origin). At level one we find the two-dimensional representation \({{\mathcal R}^{(1)}} = {{\bf{2}}_0}\) (green nodes). The black arrow denotes the negative level root −α2 and so gives rise to the level = −1 representation \({{\mathcal R}^{(-1)}} = {{\bf{2}}_(-1)}\). The blue arrows represent the fundamental weights Λ1 and Λ2.

The idea of the level decomposition is to decompose the adjoint representation into representations of one of the regular \({\mathfrak {sl}(2,\,\mathbb R)}\)-subalgebras associated with one of the two simple roots α1 or α2, i.e., either \(\{{e_1},\,\alpha _1^ \vee, \,{f_1}\}\) or \(\{{e_2},\,\,\alpha _2^ \vee, \,{f_2}\}\). For definiteness we choose the level to count the number of times the root α2 occurs, as was anticipated by the notation in Equation (8.2). Consider the subspace of the adjoint representation spanned by the vectors with a fixed value of . This subspace is invariant under the action of the subalgebra \(\{{e_1},\,\alpha _1^ \vee, \,{f_1}\}\), which only changes the value of m. Vectors at a definite level transform accordingly in a representation of the regular \({\mathfrak {sl}(2,\,\mathbb R)}\)-subalgebra
$$\mathfrak{r} \equiv \mathbb{R}{e_1} \oplus {\rm{}}\mathbb{R}\alpha _1^ \vee \oplus {\rm{}}\mathbb{R}{f_1}.$$
(8.4)
Let us begin by analyzing states at level = 0, i.e., with weights of the form γ = 1. We see from Figure 45 that we are restricted to move along the horizontal axis in the root diagram. By the defining Lie algebra relations we know that \({\rm{a}}{{\rm{d}}_{{f_1}}}({f_1}) = 0\), implying that \(\Lambda _{{\rm{ad}}}^{(0)} = - {\alpha _1}\) is a lowest weight of the \({\mathfrak {sl}(2,\,\mathbb R)}\)-representation. Here, the superscript 0 indicates that this is a level = 0 representation. The corresponding complete irreducible module is found by acting on f1 with e1, yielding
$$[{e_1},{f_1}] = \alpha _1^ \vee ,\qquad [{e_1},\alpha _1^ \vee ] = - 2{e_1},\qquad [{e_1},{e_1}] = 0.$$
(8.5)

We can then conclude that \(\Lambda _{{\rm{ad}}}^{(0)} = - {\alpha _1}\) is the lowest weight of the three-dimensional adjoint representation 30 of \({\mathfrak {sl}(2,\,\mathbb R)}\) with weights \(\{\Lambda _{{\rm{ad}}}^{(0)},\,0,\, - \Lambda _{{\rm{ad}}}^{(0)}\}\), where the subscript on 30 again indicates that this representation is located at level = 0 in the decomposition. The module for this representation is \(\mathcal L(\Lambda _{{\rm{ad}}}^{(0)}) = span\{{f_1},\,\alpha _1^ \vee, \,{e_1}\}\).

This is, however, not the complete content at level zero since we must also take into account the Cartan generator \(\alpha _2^ \vee\) which remains at the origin of the root diagram. We can combine \(\alpha _2^ \vee\) with into the vector
$$h = \alpha _1^ \vee + 2\alpha _2^ \vee ,$$
(8.6)
which constitutes the one-dimensional singlet representation 10 of \({\mathfrak r}\) since it is left invariant under all generators of \({\mathfrak r}\),
$$[{e_1},h] = [{f_1},h] = [\alpha _1^ \vee ,h] = 0,$$
(8.7)
as follows trivially from the Chevalley relations. Thus level zero contains the representations 30 and 10.

Note that the vectors at level 0 not only transform in a (reducible) representation of \({\mathfrak {sl}(2,\,\mathbb R)}\), but also form a subalgebra since the level is additive under taking commutators. The algebra in question is \({\mathfrak {gl}}(2,\,\mathbb R) = {\mathfrak {sl}}(2,\,\mathbb R) \oplus \mathbb R\). Accordingly, if the generator \(\alpha _2^ \vee\) is added to the subalgebra \({\mathfrak r}\), through the combination in Equation (8.6), so as to take the entire = 0 subspace, \({\mathfrak r}\) is enlarged from \({\mathfrak {sl}(2,\,\mathbb R)}\) to \({\mathfrak {gl}(2,\,\mathbb R)}\), the generator h being somehow the “trace” part of \({\mathfrak {gl}(2,\,\mathbb R)}\). This fact will prove to be important in subsequent sections.

Let us now ascend to the next level, = 1. The weights of \({\mathfrak r}\) at level 1 take the general form γ = 1 + α2 and the lowest weight is Λ(1) = α2, which follows from the vanishing of the commutator
$$[{f_{\rm{1}}},{e_{\rm{2}}}]{\rm{= 0}}.$$
(8.8)
Note that m ≥ 0 whenever > 0 since 1 + ℓα2 is then a positive root. The complete representation is found by acting on the lowest weight Λ(1) with e1 and we get that the commutator [e1, e2] is allowed by the Serre relations, while [e1, [e1, e2]] is killed, i.e.,
$$\begin{array}{*{20}c} {[{e_1},{e_2}] \ne 0,} \\ {\left. {[{e_1},[{e_1},{e_2}]]\,} \right] = 0.\quad \quad} \\ \end{array}$$
(8.9)
The non-vanishing commutator e θ = [e1, e2] is the vector associated with the highest root θ of \({\mathfrak {sl}(3,\,\mathbb R)}\) given by
$$\theta = {\alpha _1} + {\alpha _2}.$$
(8.10)
This is just the negative of the lowest weight \({\Lambda _\mathfrak {g}}\). The only representation at level one is thus the two-dimensional representation 21 of \(\mathfrak r\) with weights {Λ(1), θ}. The decomposition stops at level one for \({\mathfrak {sl}(3,\,\mathbb R)}\) because any commutator with two e2’s vanishes by the Serre relations. The negative level representations may be found simply by applying the Chevalley involution and the result is the same as for level one.
Hence, the total level decomposition of \({\mathfrak {sl}(3,\,\mathbb R)}\) in terms of the subalgebra \({\mathfrak {sl}(2,\,\mathbb R)}\) is given by
$$8 = {3_0} \oplus {1_0} \oplus {2_1} \oplus {2_{(- 1)}}.$$
(8.11)
Although extremely simple (and familiar), this example illustrates well the situation encountered with more involved cases below. In the following analysis we will not mention the negative levels any longer because these can always be obtained simply through a reflection with respect to the = 0 “hyperplane”, using the Chevalley involution.

8.2 Some formal considerations

Before we proceed with a more involved example, let us formalize the procedure outlined above. We mainly follow the excellent treatment given in [124], although we restrict ourselves to the cases where \(\mathfrak r\) is a finite regular subalgebra of \({\mathfrak g}\).

In the previous example, we performed the decomposition of the roots (and the ensuing decomposition of the algebra) with respect to one of the simple roots which then defined the level. In general, one may consider a similar decomposition of the roots of a rank r Kac-Moody algebra with respect to an arbitrary number s < r of the simple roots and then the level is generalized to the “multilevel” = (1, ⋯, s ).

8.2.1 Gradation

We consider a Kac-Moody algebra \({\mathfrak g}\) of rank r and we let \({\mathfrak r} \subset {\mathfrak g}\) be a finite regular rank m < r subalgebra of \({\mathfrak g}\) whose Dynkin diagram is obtained by deleting a set of nodes \({\mathcal N} = \{{n_1}, \cdots, \,{n_s}\} \,(s = r - m)\) from the Dynkin diagram of \({\mathfrak g}\).

Let γ be a root of \({\mathfrak g}\),
$$\gamma = \sum\limits_{i \notin {\mathcal N}} {{m_i}} {\alpha _i} + \sum\limits_{a \in {\mathcal N}} {{\ell _a}} {\alpha _a}.$$
(8.12)
To this decomposition of the roots corresponds a decomposition of the algebra, which is called a gradation of \({\mathfrak g}\) and which can be written formally as
$$\mathfrak{g} = \underset{\ell \in {\mathbb{Z}^s}} {\bigoplus} {\mathfrak{g}_\ell},$$
(8.13)
where for a given, \({\ell},\,{{\mathfrak g}_{\ell}}\) is the subspace spanned by all the vectors e γ with that definite value of the multilevel,
$$[h,{e_\gamma}] = \gamma (h){e_\gamma},\qquad {l_a}(\gamma) = {\ell _a}.$$
(8.14)
Of course, if \({\mathfrak g}\) is finite-dimensional this sum terminates for some finite level, as in Equation (8.11) for \({\mathfrak {sl}}(3,\,{\mathbb R})\). However, in the following we shall mainly be interested in cases where Equation (8.13) is an infinite sum.
We note for further reference that the following structure is inherited from the gradation:
$$[{\mathfrak {g}_\ell},{\mathfrak {g}_{\ell \prime}}] \subseteq {\mathfrak {g}_{\ell + \ell \prime}}.$$
(8.15)
This implies that for = 0 we have
$$[{\mathfrak {g}_0},{\mathfrak {g}_{\ell \prime}}] \subseteq {\mathfrak {g}_{\ell \prime}},$$
(8.16)
which means that \({{\mathfrak g}_{\ell}}\) is a representation of \({{\mathfrak g}_0}\) under the adjoint action. Furthermore, \({{\mathfrak g}_0}\) is a subalgebra. Now, the algebra \({\mathfrak r}\) is a subalgebra of \({{\mathfrak g}_0}\) and hence we also have
$$[\mathfrak {r}, {\mathfrak {g}_{{\ell \prime}}}] \subseteq {\mathfrak {g}_{{\ell \prime}}},$$
(8.17)
so that the subspaces \(\mathfrak {g}_\ell\) at definite values of the multilevel are invariant subspaces under the adjoint action of \({\mathfrak r}\). In other words, the action of \({\mathfrak r}\) on \({{\mathfrak g}_{\ell}}\) does not change the coefficients a .

At level zero, = (0, ⋯, 0), the representation of the subalgebra \({\mathfrak r}\) in the subspace \({{\mathfrak g}_0}\) contains the adjoint representation of \({\mathfrak r}\), just as in the case of \({\mathfrak {sl}}(3,\,{\mathbb R})\) discussed in Section 8.1. All positive and negative roots of \({\mathfrak r}\) are relevant. Level zero contains in addition s singlets for each of the Cartan generator associated to the set \({\mathcal N}\).

Whenever one of the a ’s is positive, all the other ones must be non-negative for the subspace \({{\mathfrak g}_{\ell}}\) to be nontrivial and only positive roots appear at that value of the multilevel.

8.2.2 Weights of \({\mathfrak g}\) and weights of \({\mathfrak r}\)

Let V be the module of a representation \({\mathcal R}({\mathfrak g})\) of \({\mathfrak g}\) and \(\Lambda \in {\mathfrak h}_{\mathfrak g}^{\ast}\) be one of the weights occurring in the representation. We define the action of \(h \in {{\mathfrak h}_{\mathfrak g}}\) in the representation \({\mathcal R}({\mathfrak g})\) on xV as
$$h \cdot x = \Lambda (h)x$$
(8.18)
(we consider representations of \({\mathfrak g}\) for which one can speak of “weights” [116]). Any representation of \({\mathfrak g}\) is also a representation of \({\mathfrak r}\). When restricted to the Cartan subalgebra \({{\mathfrak h}_{\mathfrak r}}\) of \({\mathfrak r}\), Λ defines a weight \(\bar \Lambda \in {\mathfrak h}_{\mathfrak r}^{\star}\), which one can realize geometrically as follows.
The dual space \({\mathfrak h}_{\mathfrak r}^{\star}\) may be viewed as the m-dimensional subspace Π of \({\mathfrak h}_{\mathfrak g}^{\star}\) spanned by the simple roots \({\alpha _i},\,i \notin {\mathcal N}\). The metric induced on that subspace is positive definite since \({\mathfrak r}\) is finite-dimensional. This implies, since we assume that the metric on \({\mathfrak h}_{\mathfrak g}^{\star}\) is nondegenerate, that \({\mathfrak h}_{\mathfrak g}^{\star}\) can be decomposed as the direct sum
$$\mathfrak {h}_\mathfrak {g}^ \star = \mathfrak {h}_\mathfrak {r}^ \star \oplus {\Pi ^ \bot}.$$
(8.19)
To that decomposition corresponds the decomposition
$$\Lambda = {\Lambda ^{\Vert}} + {\Lambda ^ \bot}$$
(8.20)
of any weight, where \({\Lambda ^\parallel} \in {\mathfrak h}_{\mathfrak r}^{\star}\) and Λ ∈ Π. Now, let \(h = \sum {{k_i}\alpha _i^ \vee \in {{\mathfrak h}_{\mathfrak r}}\,\left({i \notin {\mathcal N}} \right)}\). One has Λ(h) = Λ(h) +Λ(h) = Λ(h) because Λ(h) = 0: The component perpendicular to \({\mathfrak h}_{\mathfrak r}^{\star}\) drops out. Indeed, \({\Lambda ^ \bot}(\alpha _i^ \vee) = {{2({\Lambda ^ \bot}\vert{\alpha _i})} \over {({\alpha _i}\vert{\alpha _i})}} = 0\) for \(i \notin {\mathcal N}\).
It follows that one can identify the weight \(\bar \Lambda \in {\mathfrak h}_{\mathfrak r}^{\star}\) with the orthogonal projection \({\Lambda ^\parallel} \in {\mathfrak h}_{\mathfrak r}^{\star}\) of \(\Lambda \in {\mathfrak h}_{\mathfrak g}^{\star}\) on \({\mathfrak h}_{\mathfrak r}^{\star}\). This is true, in particular, for the fundamental weights Λ i . The fundamental weights Λ i project on 0 for \(i \in {\mathcal N}\) and project on the fundamental weights \({\bar \Lambda _i}\) of the subalgebra \({\mathfrak r}\) for \(i \notin {\mathcal N}\). These are also denoted λ i . For a general weight, one has
$$\Lambda = \sum\limits_{i \notin {\mathcal N}} {{p_i}} {\Lambda _i} + \sum\limits_{a \in {\mathcal N}} {{k_a}} {\Lambda _a}$$
(8.21)
and
$$\bar \Lambda = {\Lambda ^{\Vert}} = \sum\limits_{i \notin \mathcal N} {{p_i}{\lambda _i}.}$$
(8.22)
The coefficients p i can easily be extracted by taking the scalar product with the simple roots,
$${p_i} = {2 \over {({\alpha _i}\vert {\alpha _i})}}({\alpha _i}\vert \Lambda),$$
(8.23)
a formula that reduces to
$${p_i} = ({\alpha _i}\vert \Lambda)$$
(8.24)
in the simply-laced case. Note that (Λ ∣ Λ) > 0 even when Λ is non-spacelike.

8.2.3 Outer multiplicity

There is an interesting relationship between root multiplicities in the Kac-Moody algebra \({\mathfrak g}\) and weight multiplicites of the corresponding \({\mathfrak r}\)-weights, which we will explore here.

For finite Lie algebras, the roots always come with multiplicity one. This is in fact true also for the real roots of indefinite Kac-Moody algebras. However, as pointed out in Section 4, the imaginary roots can have arbitrarily large multiplicity. This must therefore be taken into account in the sum (8.13).

Let \(\gamma \in {\mathfrak h}_{\mathfrak g}^{\star}\) be a root of \({\mathfrak g}\). There are two important ingredients:
  • The multiplicity mult(γ) of each \(\gamma \in {\mathfrak h}_{\mathfrak g}^{\star}\) at level as a root of \({\mathfrak g}\).

  • The multiplicity mult\(_{{\mathcal R}_\gamma ^(\ell)}(\gamma)\) of the corresponding weight \(\bar \gamma \in {\mathfrak h}_{\mathfrak r}^{\star}\) at level as a weight in the representation \({\mathcal R}_\gamma ^{(\ell)}\) of \({\mathfrak r}\). (Note that two distinct roots at the same level project on two distinct \({\mathfrak r}\)-weights, so that given the \({\mathfrak r}\)-weight and the level, one can reconstruct the root.)

It follows that the root multiplicity of γ is given as a sum over its multiplicities as a weight in the various representations \({\mathcal R}_\gamma ^{(\ell)}\vert q = 1, \cdots, {{\mathcal N}_{\ell}}\) at level . Some representations can appear more than once at each level, and it is therefore convenient to introduce a new measure of multiplicity, called the outer multiplicity \(\mu ({\mathcal R}_q ^{(\ell)})\), which counts the number of times each representation \(({\mathcal R}_q ^{(\ell)})\) appears at level . So, for each representation at level we must count the individual weight multiplicities in that representation and also the number of times this representation occurs. The total multiplicity of γ can then be written as
$${\rm{mult}}(\gamma) = \sum\limits_{q = 1}^{{N_\ell}} \mu ({\mathcal R}_q^{(\ell)}){\rm{mul}}{{\rm{t}}_{{\mathcal R}_q^{(\ell)}}}(\gamma)$$
(8.25)
This simple formula might provide useful information on which representations of \({\mathfrak r}\) are allowed within \({\mathfrak g}\) at a given level. For example, if γ is a real root of \({\mathfrak g}\), then it has multiplicity one. This means that in the formula (8.25), only the representations of \({\mathfrak r}\) for which γ has weight multiplicity equal to one are permitted. The others have \(\mu ({\mathcal R}_q^{(\ell)}) = 0\). Furthermore, only one of the permitted representations does actually occur and it has necessarily outer multiplicity equal to one, \(\mu ({\mathcal R}_q^{(\ell)}) = 1\).
The subspaces \({\mathfrak g}_{\ell}\) can now be written explicitly as
$${\mathfrak{g}_\ell } = \mathop {\mathop \oplus \limits_{q = 1} }\limits^{{N_\ell }} \left[ {\mathop {\mathop \oplus \limits_{k = 1} }\limits^{\mu (\mathcal{R}_q^{(\ell )})} {\mathcal{L}^{[k]}}(\Lambda _q^{(\ell )})} \right],$$
(8.26)
where \({\mathcal L}(\Lambda _q^{(\ell)})\) denotes the module of the representation \({\mathcal R}_q ^{(\ell)}\) and N is the number of inequivalent representations at level í. It is understood that if \(\mu ({\mathcal R}_q^{(\ell)}) = 0\) for some and q, then \({\mathcal L}(\Lambda _q^{(\ell)})\) is absent from the sum. Note that the superscript [k] labels multiple modules associated to the same representation, e.g., if \(\mu ({\mathcal R}_q^{(\ell)}) = 3\) this contributes to the sum with a term
$${{\mathcal L}^{[1]}}(\Lambda _q^{(\ell)}) \oplus {{\mathcal L}^{[2]}}(\Lambda _q^{(\ell)}) \oplus {{\mathcal L}^{[3]}}(\Lambda _q^{(\ell)}){.}$$
(8.27)
Finally, we mention that the multiplicity mult(α) of a root \(\alpha \in {{\mathfrak h}^{\star}}\) can be computed recursively using the Peterson recursion relation, defined as [116]
$$(\alpha \vert \alpha - 2\,\rho){c_\alpha} = \sum\limits_{{\underset{\gamma ,\gamma \prime \,\, \in \,\,Q + \,\,\,} {\gamma + \gamma \prime = \alpha}}} {(\gamma \vert \gamma \prime){c_\gamma}{c_{\gamma \prime}},}$$
(8.28)
where Q+ denotes the set of all positive integer linear combinations of the simple roots, i.e., the positive part of the root lattice, and ρ is the Weyl vector (defined in Section 4). The coefficients c γ are defined as
$${c_\gamma} = \sum\limits_{k \ge 1} {{1 \over k}} \,{\rm{mult}}\left({{\gamma \over k}} \right),$$
(8.29)
and, following [19], we call this the co-multiplicity. Note that if γ/k is not a root, this gives no contribution to the co-multiplicity. Another feature of the co-multiplicity is that even if the multiplicity of some root γ is zero, the associated co-multiplicity c γ does not necessarily vanish. Taking advantage of the fact that all real roots have multiplicity one it is possible, in principle, to compute recursively the multiplicity of any imaginary root. Since no closed formula exists for the outer multiplicity μ, one must take a detour via the Peterson relation and Equation (8.25) in order to find the outer multiplicity of each representation at a given level. We give in Table 38 a list of root multiplicities and co-multiplicities of roots of AE3 up to height 10.
Table 38

Multiplicities m α = mult(α) and co-multiplicities c α of all roots α of AE3 up to height 10.

m 1

m 2

c α

m α

α 2

0

0

1

1

1

2

0

0

k > 1

1/k

0

2k2

0

1

0

1

1

2

0

k > 1

1

1/k

0

2k2

1

0

0

1

1

2

k > 0

0

0

1/k

0

2 k2

0

1

1

1

1

2

0

k > 1

k > 1

1/k

0

2 k2

1

1

0

1

1

0

2

2

0

3/2

1

0

3

3

0

4/3

1

0

4

4

0

7/4

1

0

5

5

0

6/5

1

0

1

1

1

1

1

0

2

2

2

3/2

1

0

3

3

3

4/3

1

0

1

2

0

1

1

2

2

4

0

1/2

0

8

3

6

0

1/3

0

2

2

1

0

1

1

2

4

2

0

1/2

0

8

6

3

0

1/3

0

18

1

2

1

1

1

0

2

4

2

3/2

1

0

2

1

1

1

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8.3 Level decomposition of AE3

The Kac-Moody algebra \(A{E_3} = A_1^{+ +}\) is one of the simplest hyperbolic algebras and so provides a nice testing ground for investigating general properties of hyperbolic Kac-Moody algebras. From a physical point of view, it is the Weyl group of AE3 which governs the chaotic behavior of pure four-dimensional gravity close to a spacelike singularity [46], as we have explained. Moreover, as we saw in Section 3, the Weyl group of AE3 is isomorphic with the well-known arithmetic group \(PGL(2,\,{\mathbb Z})\) which has interesting properties [75].

The level decomposition of \({\mathfrak g} = A{E_3}\) follows a similar route as for \({\mathfrak {sl}}(3,\,{\mathbb R})\) above, but the result is much more complicated due to the fact that AE3 is infinite-dimensional. This decomposition has been treated before in [48]. Recall that the Cartan matrix for AE3 is given by
$$\left({\begin{array}{*{20}c} 2 & {- 2\,\,\,} & 0 \\ {- 2\,\,\,} & 2 & {- 1\,\,\,} \\ 0 & {- 1\,\,\,} & 2 \\ \end{array}} \right)$$
(8.30)
and the associated Dynkin diagram is given in Figure 46.
Figure 46

The Dynkin diagram of the hyperbolic Kac-Moody algebra \(A{E_3} \equiv A_1^{+ +}\). The labels indicate the simple roots α1,α2 and α3. The nodes “2” and “3” correspond to the subalgebra \({\mathfrak r} = {\mathfrak {sl}}(3,\,{\mathbb R})\) with respect to which we perform the level decomposition.

We see that there exist three rank 2 regular subalgebras that we can use for the decomposition: \({A_2},\,{A_1} \oplus {A_1}\) or \(A_1^ +\). We will here focus on the decomposition into representations of \({\mathfrak r} = {A_2} = {\mathfrak {sl}}(3,\,{\mathbb R})\) because this is the one relevant for pure gravity in four dimensions [46]31. The level is then the coefficient in front of the simple root α1 in an expansion of an arbitrary root \(\gamma \in {\mathfrak h}_{\mathfrak g}^{\star}\), i.e.,
$$\gamma = \ell {\alpha _1} + {m_2}{\alpha _2} + {m_3}{\alpha _3}.$$
(8.31)
We restrict henceforth our analysis to positive levels only, ≥ 0. Before we begin, let us develop an intuitive idea of what to expect. We know that at each level we will have a set of finite-dimensional representations of the subalgebra \({\mathfrak r}\). The corresponding weight diagrams will then be represented in a Euclidean two-dimensional lattice in exactly the same way as in Figure 45 above. The level can be understood as parametrizing a third direction that takes us into the full three-dimensional root space of AE3. We display the level decomposition up to positive level two in Figure 4732.
Figure 47

Level decomposition of the adjoint representation of AE3. We have displayed the decomposition up to positive level = 2. At level zero we have the adjoint representation \({\mathcal R}_1^{(0)} = {8_0}\) of \({\mathfrak {sl}}(3,\,{\mathbb R})\) and the singlet representation \({\mathcal R}_2^{(0)} = {1_0}\) defined by the simple Cartan generator \(\alpha _1^ \vee\). Ascending to level one with the root α1 (green vector) gives the lowest weight Λ(1) of the representation \({{\mathcal R}^{(1)}} = {6_1}\). The weights of \({{\mathcal R}^{(1)}}\) labelled by white crosses are on the lightcone and so their norm squared is zero. At level two we find the lowest weight Λ(2) (blue vector) of the 15-dimensional representation \({{\mathcal R}^{(2)}} = {15_2}\). Again, the white crosses label weights that are on the lightcone. The three innermost weights are inside of the lightcone and the rings indicate that these all have multiplicity 2 as weights of \({{\mathcal R}^{(2)}}\). Since these also have multiplicity 2 as roots of \({\mathfrak h}_{\mathfrak g}^{\star}\) we find that the outer multiplicity of this representation is one, \(\mu ({\mathcal R}^{(2)}) = 1\).

From previous sections we recall that AE3 is hyperbolic so its root space is of Lorentzian signature. This implies that there is a lightcone in \({\mathfrak h}_{\mathfrak g}^{\star}\) whose origin lies at the origin of the root diagram for the adjoint representation of \({\mathfrak r}\) at level = 0. The lightcone separates real roots from imaginary roots and so it is clear that if a representation at some level intersects the walls of the lightcone, this means that some weights in the representation will correspond to imaginary roots of \({\mathfrak h}_{\mathfrak g}^{\star}\) but will be real as weights of \({\mathfrak h}_{\mathfrak r}^{\star}\). On the other hand if a weight lies outside of the lightcone it will be real both as a root of \({\mathfrak h}_{\mathfrak g}^{\star}\) and as a weight of \({\mathfrak h}_{\mathfrak r}^{\star}\).

8.3.1 Level = 0

Consider first the representation content at level zero. Given our previous analysis we expect to find the adjoint representation of \({\mathfrak r}\) with the additional singlet representation from the Cartan generator \(\alpha _1^ \vee\). The Chevalley generators of \({\mathfrak r}\) are \(\{{e_2},\,{f_2},\,{e_3},\,{f_3},\,\alpha _2^ \vee, \,\alpha _3^ \vee \}\) and the generators associated to the root defining the level are \(\{{e_1},\,{f_1},\,\alpha _1^ \vee \}\). As discussed previously, the additional Cartan generator \(\alpha _1^ \vee\) that sits at the origin of the root space enlarges the subalgebra from \({\mathfrak {sl}}(3,\,{\mathbb R})\) to \({\mathfrak {gl}}(3,\,{\mathbb R})\). A canonical realisation of \({\mathfrak {gl}}(3,\,{\mathbb R})\) is obtained by defining the Chevalley generators in terms of the matrices \({K^i}_j{\rm{(}}i{\rm{,}}j{\rm{= 1, 2, 3)}}\) whose commutation relations are
$$[{K^i}_j,{K^k}_l] = \delta _j^k{K^i}_l - \delta _l^i{K^k}_j.$$
(8.32)
All the defining Lie algebra relations of \({\mathfrak {gl}}(3,\,{\mathbb R})\) are then satisfied if we make the identifications
$$\begin{array}{*{20}c} {\qquad \quad \quad \quad \quad \quad \quad \quad \quad \alpha _1^ \vee = {K^1}_1 - {K^2}_2 - {K^3}_3,\,\,\,} \\ {{e_2} = {K^2}_1,\quad \quad {f_2} = {K^1}_2,\quad \quad \alpha _2^ \vee = {K^2}_2 - {K^1}_1,\quad \quad \quad \quad} \\ {{e_3} = {K^3}_2,\quad \quad {f_3} = {K^2}_3,\quad \quad \alpha _3^ \vee = {K^3}_3 - {K^2}_2.\quad \quad \quad \quad} \\ \end{array}$$
(8.33)
Note that the trace \({k^1}_1 + {k^2}_2 + {k^3}_3\) is equal to \(- 4\alpha _2^ \vee - 2\alpha _3^ \vee - 3\alpha _1^ \vee\). The generators e1 and f1 can of course not be realized in terms of the matrices \({K^i}_j\) since they do not belong to level zero. The invariant bilinear form (|) at level zero reads
$$({K^i}_j\vert {K^k}_l) = \delta _l^i\delta _j^k - \delta _j^i\delta _l^k,$$
(8.34)
where the coefficient in front of the second term on the right hand side is fixed to −1 through the embedding of \({\mathfrak {gl}}(3,\,{\mathbb R})\) in AE3.

The commutation relations in Equation (8.32) characterize the adjoint representation of \({\mathfrak {gl}}(3,\,{\mathbb R})\) as was expected at level zero, which decomposes as the representation \({\mathcal R}_{{\rm{ab}}}^{(0)} \oplus {\mathcal R}_s^{(0)}\) of \({\mathfrak {sl}}(3,\,{\mathbb R})\) with \({\mathcal R}_{{\rm{ab}}}^{(0)} = {8_0}\) and \({\mathcal R}_{{\rm s}}^{(0)} = {1_0}\).

8.3.2 Dynkin labels

It turns out that at each positive level , the weight that is easiest to identify is the lowest weight. For example, at level one, the lowest weight is simply α1 from which one builds all the other weights by adding appropriate positive combinations of the roots α2 and α3. It will therefore turn out to be convenient to characterize the representations at each level by their (conjugate) Dynkin labels p2 and p3 defined as the coefficients of minus the (projected) lowest weight \(- \bar \Lambda _{{\mathrm{lw}}}^{(\ell)}\) expanded in terms of the fundamental weights λ2 and λ3 of \(\mathfrak{sl}(3,\, \mathbb{R})\) (blue arrows in Figure 48),
$$- \bar \Lambda _{{\rm{lw}}}^{(\ell)} = {p_2}{\lambda _2} + {p_3}{\lambda _3}.$$
(8.35)
Note that for any weight Λ we have the inequality
$$(\Lambda \vert \Lambda) \le (\bar \Lambda \vert \bar \Lambda)$$
(8.36)
since \((\Lambda \vert\Lambda) = (\bar \Lambda \vert\bar \Lambda) - \vert({\Lambda ^ \bot}\vert{\Lambda ^ \bot})\vert\).
Figure 48

The representation 152 of \(\mathfrak{sl}(3,\, \mathbb{R})\) appearing at level two in the decomposition of the adjoint representation of AE3 into representations of \(\mathfrak{sl}(3,\, \mathbb{R})\). The lowest leftmost node is the lowest weight of the representation, corresponding to the real root Λ(2) = 2α1 + α2 of AE3. This representation has outer multiplicity one.

The Dynkin labels can be computed using the scalar product (|) in \(\mathfrak{h}_{\mathfrak{g}}^{\star}\) in the following way:
$${p_2} = - ({\alpha _2}\vert \Lambda _{{\rm{lw}}}^{(\ell)}),\qquad {p_3} = - ({\alpha _3}\vert \Lambda _{{\rm{lw}}}^{(\ell)}){.}$$
(8.37)
for the level zero sector we therefore have
$$\begin{array}{*{20}c} {{{\bf{8}}_0}:[{p_2},{p_3}] = [1,1],} \\ {{{\bf{1}}_0}:[{p_2},{p_3}] = [0,0].} \\ \end{array}$$
(8.38)

The module for the representation 80 is realized by the eight traceless generators \({K^i}_j\) of \(\mathfrak{sl}(3,\, \mathbb{R})\) and the module for the representation 10 corresponds to the “trace” \(\alpha _1^ \vee\).

Note that the highest weight Λhw of a given representation of \(\mathfrak{r}\) is not in general equal to minus the lowest weight Λ of the same representation. In fact, −Λhw is equal to the lowest weight of the conjugate representation. This is the reason our Dynkin labels are really the conjugate Dynkin labels in standard conventions. It is only if the representation is self-conjugate that we have Λhw = −Λ. This is the case for example in the adjoint representation 80.

It is interesting to note that since the weights of a representation at level are related by Weyl reflections to weights of a representation at level −, it follows that the negative of a lowest weight Λ(ℓ) at level is actually equal to the highest weight \(\Lambda _{{\mathrm{hw}}}^{(- \ell)}\) of the conjugate representation at level −. Therefore, the Dynkin labels at level as defined here are the standard Dynkin labels of the representations at level −.

8.3.3 Level = 1

We now want to exhibit the representation content at the next level =1. A generic level one commutator is of the form [e1, [⋯ [⋯]]], where the ellipses denote (positive) level zero generators. Hence, including the generator e1 implies that we step upwards in root space, i.e., in the direction of the forward lightcone. The root vector e1 corresponds to a lowest weight of \(\mathfrak{r}\) since it is annihilated by f2 and f3,
$$\begin{array}{*{20}c} {{\rm{a}}{{\rm{d}}_{{f_2}}}({e_1}) = [{f_2},{e_1}] = 0,} \\ {{\rm{a}}{{\rm{d}}_{{f_3}}}({e_1}) = [{f_{3,}}{e_1}] = 0,} \\ \end{array}$$
(8.39)
which follows from the defining relations of AE3.
Explicitly, the root associated to e1 is simply the root α1 that defines the level expansion. Therefore the lowest weight of this level one representation is
$$\Lambda _{{\rm{lw}}}^{(1)} = {\bar \alpha _1},$$
(8.40)
Although α1 is a real positive root of \(\mathfrak{h}_{\mathfrak{g}}^{\star}\), its projection \({\bar \alpha _{(1)}}\) is a negative weight of \(\mathfrak{h}_{\mathfrak{r}}^{\star}\). Note that since the lowest weight \(\Lambda _1^{(1)}\) is real, the representation \({{\mathcal R}^{(1)}}\) has outer multiplicity one, \(\mu ({{\mathcal R}^{(1)}}) = 1\).
Acting on the lowest weight state with the raising operators of \(\mathfrak{r}\) yields the six-dimensional representation \({{\mathcal R}^{(1)}} = {{\mathbf{6}}_1}\) of \(\mathfrak{sl}(3,\, \mathbb{R})\). The root α1 is displayed as the green vector in Figure 47, taking us from the origin at level zero to the lowest weight of The Dynkin labels of this representation are
$$\begin{array}{*{20}c} {{p_2}({{\mathcal R}^{(1)}}) = - ({\alpha _2}\vert {\alpha _1}) = 2,} \\ {{p_3}({{\mathcal R}^{(1)}}) = - ({\alpha _3}\vert {\alpha _1}) = 0,} \\ \end{array}$$
(8.41)
which follows directly from the Cartan matrix of AE3. Three of the weights in correspond to roots that are located on the lightcone in root space and so are null roots of \(\mathfrak{h}_{\mathfrak{g}}^{\star}\). These are α1 + α2, α1 + α2 + α3 and α1 + 2α2 + α3 and are labelled with white crosses in Figure 47. The other roots present in the representation, in addition to α1, are α1 + 2α2 and α1 + 2α2 + 2α3, which are real. This representation therefore contains no weights inside the lightcone.
The \(\mathfrak{gl}(3,\, \mathbb{R})\)-generator encoding this representation is realized as a symmetric 2-index tensor E ij which indeed carries six independent components. In general we can easily compute the dimensionality of a representation given its Dynkin labels using the Weyl dimension formula which for \(\mathfrak{sl}(3,\, \mathbb{R})\) takes the form [84]
$${d_{{\Lambda _{{\rm{hw}}}}}}\left({\mathfrak sl (3,\mathbb R)} \right) = ({p_2} + 1)({p_3} + 1)\left({{1 \over 2}({p_2} + {p_3}) + 1} \right).$$
(8.42)
In particuar, for (p2,p3) = (2, 0) this gives indeed \({d_{\Lambda _{{\mathrm{hw}},\,1}^{(1)}}} = 6\).
It is convenient to encode the Dynkin labels, and, consequently, the index structure of a given representation module, in a Young tableau. We follow conventions where the first Dynkin label gives the number of columns with 1 box and the second Dynkin label gives the number of columns with 2 boxes33. For the representation 61 the first Dynkin label is 2 and the second is 0, hence the associated Young tableau is
At level = −1 there is a corresponding negative generator F ij . The generators E ij and F ij transform contravariantly and covariantly, respectively, under the level zero generators, i.e.,
$$\begin{array}{*{20}c} {[{K^i}_j,{E^{kl}}] = \delta _j^k{E^{il}} + \delta _j^l{E^{ki}},} \\ {[{K^i}_j,{F_{kl}}] = - \delta _k^i{F_{jl}} - \delta _l^i{F_{kj}}.} \\ \end{array}$$
(8.44)
The internal commutator on level one can be obtained by first identifying
$${e_1} \equiv {E^{11}},\qquad {f_1} \equiv {F_{11}},$$
(8.45)
and then by demanding [e1, f1] = we find
$$[{E^{ij}},\,{F_{kl}}] = 2\delta _{(k}^{(i}{K^{j)}}_{l)} - \delta _k^{(i}\delta _l^{k)}({K^1}_1 + {K^2}_2 + {K^3}_3),$$
(8.46)
which is indeed compatible with the realisation of given in Equation (8.33). The Killing form at level 1 takes the form
$$\left({{F_{ij}}\vert {E^{kl}}} \right) = \delta _i^{(k}\delta _j^{l)}.$$
(8.47)

8.3.4 Constraints on Dynkin labels

As we go to higher and higher levels it is useful to employ a systematic method to investigate the representation content. It turns out that it is possible to derive a set of equations whose solutions give the Dynkin labels for the representations at each level [47].

We begin by relating the Dynkin labels to the expansion coefficients , m2 and m3 of a root \(\gamma \in \mathfrak{h}_{\mathfrak{g}}^{\star}\), whose projection \(\bar \gamma\) onto \(\mathfrak{h}_{\mathfrak{r}}^{\star}\) is a lowest weight vector for some representation of \(\mathfrak{r}\) at level . We let α = 2, 3 denote indices in the root space of the subalgebra \(\mathfrak{sl}(3,\, \mathbb{R})\) and we let i = 1, 2, 3 denote indices in the full root space of AE3. The formula for the Dynkin labels then gives
$${p_a} = - ({\alpha _a}\vert \gamma) = - \ell {A_{a1}} - {m_2}{A_{a2}} - {m_3}{A_{a3}},$$
(8.48)
where A ij is the Cartan matrix for AE3, given in Equation (8.30). Explicitly, we find the following relations between the coefficients m2,m3 and the Dynkin labels:
$$\begin{array}{*{20}c} {{p_2} = 2\ell - 2{m_2} + {m_3},} \\ {{p_{\rm{3}}}{\rm{=}}{m_{\rm{2}}} - 2{m_{\rm{3}}}{\rm{.}}\quad \quad} \\ \end{array}$$
(8.49)
These formulae restrict the possible Dynkin labels for each since the coefficients m2 and m3 must necessarily be non-negative integers. Therefore, by inverting Equation (8.49) we obtain two Diophantine equations that restrict the possible Dynkin labels,
$$\begin{array}{*{20}c} {{m_2} = {4 \over 3}\ell - {2 \over 3}{p_2} - {1 \over 3}{p_3} \ge 0,} \\ {{m_3} = {2 \over 3}\ell - {1 \over 3}{p_2} - {2 \over 3}{p_3} \ge 0.} \\ \end{array}$$
(8.50)
In addition to these constraints we can also make use of the fact that we are decomposing the adjoint representation of AE3. Since the weights of the adjoint representation are the roots of the algebra we know that the lowest weight vector Λ must satisfy
$$(\Lambda \vert \Lambda) \le 2.$$
(8.51)
Taking Λ = ℓα1 + m2α2 + m3α3 then gives the following constraint on the coefficients , m2 and m3:
$$(\Lambda \vert \Lambda) = 2{\ell ^2} + 2m_2^2 + 2m_3^2 - 4\ell {m_2} - 2{m_2}{m_3} \le 2.$$
(8.52)
We are interested in finding an equation for the Dynkin labels, so we insert Equation (8.50) into Equation (8.52) to obtain the constraint
$$p_2^2 + p_3^2 + {p_2}{p_3} - {\ell ^2} \le 3.$$
(8.53)
The inequalities in Equation (8.50) and Equation (8.53) are sufficient to determine the representation content at each level . However, this analysis does not take into account the outer multiplicities, which must be analyzed separately by comparing with the known root multiplicities of AE3 as given in Table 38. We shall return to this issue later.

8.3.5 Level = 2

Let us now use these results to analyze the case for which = 2. The following equations must then be satisfied:
$$\begin{array}{*{20}c} {a8 - 2{p_2} - {p_3} \ge 0,} \\ {\,\,4 - {p_2} - 2{p_3} \ge 0,} \\ {p_2^2 + p_3^2 + {p_2}{p_3} \le 7.} \\ \end{array}$$
(8.54)
The only admissible solution is p2 = 2 and p3 = 1. This corresponds to a 15-dimensional representation 152 with the following Young tableau Note that p2 = p3 = 0 is also a solution to Equation (8.54) but this violates the constraint that m2 and m3 be integers and so is not allowed.

Moreover, the representation [p2,p3] = [0, 2] is also a solution to Equation (8.54) but has not been taken into account because it has vanishing outer multiplicity. This can be understood by examining Figure 48 a little closer. The representation [0, 2] is six-dimensional and has highest weight 2λ3, corresponding to the middle node of the top horizontal line in Figure 48. This weight lies outside of the lightcone and so is a real root of AE3. Therefore we know that it has root multiplicity one and may therefore only occur once in the level decomposition. Since the weight 2λ3 already appears in the larger representation 152 it cannot be a highest weight in another representation at this level. Hence, the representation [0, 2] is not allowed within AE3. A similar analysis reveals that also the representation [p2,p3] = [1, 0], although allowed by Equation (8.54), has vanishing outer multiplicity.

The level two module is realized by the tensor \(E_i^{\,jk}\) whose index structure matches the Young tableau above. Here we have used the \(\mathfrak{sl}(3,\, \mathbb{R})\)-invariant antisymmetric tensor abc to lower the two upper antisymmetric indices leading to a tensor \(E_i^{\,jk}\) with the properties
$${E_i}^{jk} = {E_i}^{(jk)},\qquad {E_i}^{ik} = 0.$$
(8.56)
This corresponds to a positive root generator and by the Chevalley involution we have an associated negative root generator \({F^i}_{jk}\) at level = −2. Because the level decomposition gives a gradation of AE3 we know that all higher level generators can be obtained through commutators of the level one generators. More specifically, the level two tensor \(E_i^{\,jk}\) corresponds to the commutator
$$[{E^{ij}},{E^{kl}}] = {\epsilon ^{mk(i}}{E_m}^{j)l} + {\epsilon ^{ml(i}}{E_m}^{j)k},$$
(8.57)
where ijk is the totally antisymmetric tensor in three dimensions. Inserting the result p2 =2 and p3 = 1 into Equation (8.50) gives m2 = 1 and m3 = 0, thus providing the explicit form of the root taking us from the origin of the root diagram in Figure 47 to the lowest weight of 152 at level two:
$${\Lambda ^{(2)}} = 2{\alpha _1} + {\alpha _2}.$$
(8.58)
This is a real root of AE3, (γ|γ) = 2, and hence the representation 152 has outer multiplicity one. We display the representation 152 of \(\mathfrak{sl}(3,\, \mathbb{R})\) in Figure 48. The lower leftmost weight is the lowest weight Λ(2). The expansion of the lowest weight \(\Lambda _{{\mathrm{lw}}}^{(2)}\) in terms of the fundamental weights λ2 and λ3 is given by the (conjugate) Dynkin labels
$$- \Lambda _{{\rm{hw}}}^{(2)} = {p_2}{\lambda _2} + {p_3}{\lambda _3} = 2{\lambda _2} + {\lambda _3}.$$
(8.59)
The three innermost weights all have multiplicity 2 as weights of \(\mathfrak{sl}(3,\, \mathbb{R})\), as indicated by the black circles. These lie inside the lightcone of \(\mathfrak{h}_{\mathfrak{g}}^{\star}\) and so are timelike roots of AE3.

8.3.6 Level = 3

We proceed quickly past level three since the analysis does not involve any new ingredients. Solving Equation (8.50) and Equation (8.53) for = 3 yields two admissible \(\mathfrak{sl}(3,\, \mathbb{R})\) representations, 273 and 83, represented by the following Dynkin labels and Young tableaux: The lowest weight vectors for these representations are
$$\begin{array}{*{20}c} {\Lambda _{{\bf{15}}}^{(3)} = 3{\alpha _1} + 2{\alpha _2},\quad \,\,\,\,} \\ {\Lambda _{\bf{8}}^{(3)} = 3{\alpha _1} + 3{\alpha _2} + {\alpha _3}.} \\ \end{array}$$
(8.61)
The lowest weight vector for 273 is a real root of AE3, \((\Lambda _{{\mathbf{27}}}^{(3)}\vert\Lambda _{{\mathbf{27}}}^{(3)}) = 2\), while the lowest weight vectors for 83 is timelike, \((\Lambda _{\mathbf{8}}^{(3)}\vert\Lambda _{\mathbf{8}}^{(3)}) = - 4\). This implies that the entire representation 83 lies inside the lightcone of \(\mathfrak{h}_{\mathfrak{g}}^{\star}\). Both representations have outer multiplicity one.

Note that [0, 3] and [3, 0] are also admissible solutions but have vanishing outer multiplicities by the same arguments as for the representation [0, 2] at level 2.

8.3.7 Level = 4

At this level we encounter for the first time a representation with non-trivial outer multiplicity. It is a 15-dimensional representation with the following Young tableau structure: The lowest weight vector is
$$\Lambda _{\overline {{\bf{15}}}}^{(4)} = 4{\alpha _1} + 4{\alpha _2} + {\alpha _3},$$
(8.63)
which is an imaginary root of AE3,
$$(\Lambda _{\overline {{\bf{15}}}}^{(4)}\vert \Lambda _{\overline {{\bf{15}}}}^{(4)}) = - 6{.}$$
(8.64)
From Table 38 we find that this root has multiplicity 5 as a root of AE3,
$${\rm{mult}}(\Lambda _{\overline {{\bf{15}}}}^{(4)}) = 5{.}$$
(8.65)
In order for Equation (8.26) to make sense, this multiplicity must be matched by the total multiplicity of \(\Lambda_{\bar{\mathbf{15}}}^{(4)}\) as a weight of \(\mathfrak{sl}(3,\, \mathbb{R})\) representations at level four. The remaining representations at this level are By drawing these representations explicitly, one sees that the root 4α1 + 4α2 + α3, representing the weight \(\Lambda_{\bar{\mathbf{15}}}^{(4)}\), also appears as a weight (but not as a lowest weight) in the representations 424 and 244. It occurs with weight multiplicity 1 in the 244 but with weight multiplicity 2 in the 424. Taking also into account the representation \({\bar{\mathbf{15}}}_{4}\) in which it is the lowest weight we find a total weight multiplicity of 4. This implies that, since in AE3
$${\rm{mult}}(4{\alpha _1} + 4{\alpha _2} + {\alpha _3}) = 5,$$
(8.67)
the outer multiplicity of \({\bar{\mathbf{15}}}_{4}\) must be 2, i.e.,
$$\mu \left({\Lambda _{\overline {{\bf{15}}}}^{(2)}} \right) = 2{.}$$
(8.68)
When we go to higher and higher levels, the outer multiplicities of the representations located entirely inside the lightcone in \(\mathfrak{h}_{\mathfrak{g}}\) increase exponentially.

8.4 Level decomposition of E10

As we have seen, the Kac-Moody algebra E10 is one of the four hyperbolic algebras of maximal rank, the others being BE10, DE10 and CE10. It can be constructed as an overextension of E8 and is therefore often denoted by \(E_8^{+ +}\). Similarly to E8 in the rank 8 case, E10 is the unique indefinite rank 10 algebra with an even self-dual root lattice, namely the Lorentzian lattice Π1,9.

Our first encounter with E10 in a physical application was in Section 5 where we have showed that the Weyl group of E10 describes the chaos that emerges when studying eleven-dimensional supergravity close to a spacelike singularity [45].

In Section 9.3, we will discuss how to construct a Lagrangian manifestly invariant under global E10-transformations and compare its dynamics to that of eleven-dimensional supergravity. The level decomposition associated with the removal of the “exceptional node” labelled “10” in Figure 49 will be central to the analysis. It turns out that the low-level structure in this decomposition precisely reproduces the bosonic field content of eleven-dimensional supergravity [47].
Figure 49

The Dynkin diagram of E10. Labels i = 1, ⋯, 9 enumerate the nodes corresponding to simple roots α i of the \(\mathfrak{sl}(10,\, \mathbb{R})\) subalgebra and “10” labels the exceptional node.

Moreover, decomposing E10 with respect to different regular subalgebras reproduces also the bosonic field contents of the Type IIA and Type IIB supergravities. The fields of the IIA theory are obtained by decomposition in terms of representations of the \(D_{9}=\mathfrak{so}(9,\,9,\, \mathbb{R})\) subalgebra obtained by removing the first simple root α1 [125]. Similarly the IIB-fields appear at low levels for a decomposition with respect to the \(A_{9} \oplus A_{1}=\mathfrak{sl}(9,\, \mathbb{R})\, \oplus \, \mathfrak{sl}(2,\, \mathbb{R})\) subalgebra found upon removal of the second simple root α2 [126]. The extra A1-factor in this decomposition ensures that the \(SL(2,\, \mathbb{R})\)-symmetry of IIB supergravity is recovered.

For these reasons, we investigate now these various level decompositions.

8.4.1 Decomposition with respect to \(\mathfrak{sl}(10,\, \mathbb{R})\)

Let α1, ⋯, α10 denote the simple roots of E10 and \(\alpha _1^ \vee, \, \cdots \,,\,\alpha _{10}^ \vee\) the Cartan generators. These span the root space \(\mathfrak{h}^{\star}\) and the Cartan subalgebra \(\mathfrak{h}\), respectively. Since E10 is simply laced the Cartan matrix is given by the scalar products between the simple roots:
$${A_{ij}}[{E_{10}}] = ({\alpha _i}\vert {\alpha _j}) = \left({\begin{array}{*{20}c} 2 & {- 1\,\,\,} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {- 1\,\,\,} & 2 & {- 1\,\,\,} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {- 1\,\,\,} & 2 & {- 1\,\,\,} & 0 & 0 & 0 & 0 & 0 & {- 1\,\,\,} \\ 0 & 0 & {- 1\,\,\,} & 2 & {- 1\,\,\,} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {- 1\,\,\,} & 2 & {- 1\,\,\,} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {- 1\,\,\,} & 2 & {- 1\,\,\,} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {- 1\,\,\,} & 2 & {- 1\,\,\,} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {- 1\,\,\,} & 2 & {- 1\,\,\,} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {- 1\,\,\,} & 2 & 0 \\ 0 & 0 & {- 1\,\,\,} & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\ \end{array}} \right)$$
(8.69)
The associated Dynkin diagram is displayed in Figure 49. We will perform the decomposition with respect to the \(\mathfrak{sl}(10,\, \mathbb{R})\) subalgebra represented by the horizontal line in the Dynkin diagram so the level of an arbitrary root \(\alpha \, \in \,\mathfrak{h}^{\star}\) is given by the coefficient in front of the exceptional simple root, i.e.,
$$\gamma = \sum\limits_{i = 1}^9 {{m^i}} {\alpha _i} + \ell {\alpha _{10}}.$$
(8.70)
As before, the weight that is easiest to identify for each representation at positive level ί is the lowest weight \(\Lambda _{{\mathrm{lw}}}^{(\ell)}\). We denote by \(\bar \Lambda _{{\mathrm{lw}}}^{(\ell)}\) the projection onto the spacelike slice of the root lattice defined by the level . The (conjugate) Dynkin labels p1, ⋯,p9 characterizing the representation \({\mathcal R}({\Lambda ^{(\ell)}})\) are defined as before as minus the coefficients in the expansion of \(\bar \Lambda _{{\mathrm{lw}}}^{(\ell)}\) in terms of the fundamental weights λ i of \(\mathfrak{sl}(10,\, \mathbb{R})\):
$$- \bar \Lambda _{{\rm{lw}}}^{(\ell)} = \sum\limits_{i = 1}^9 {{p_i}} {\lambda ^i}.$$
(8.71)
The Killing form on each such slice is positive definite so the projected weight \(\bar \Lambda _{{\mathrm{lw}}}^{(\ell)}\) of course real. The fundamental weights of \(\mathfrak{sl}(10,\, \mathbb{R})\) can be computed explicitly from their definition as the duals of the simple roots:
$${\lambda ^i} = \sum\limits_{j = 1}^9 {{B^{ij}}} {\alpha _j},$$
(8.72)
where B ij is the inverse of the Cartan matrix of A9,
$${\left({{B_{ij}}[{A_9}]} \right)^{- 1}} = {1 \over {10}}\left({\begin{array}{*{20}c} 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\ 8 & {16} & {14} & {12} & {10} & 8 & 6 & 4 & 2 \\ 7 & {14} & {21} & {18} & {15} & {12} & 9 & 6 & 3 \\ 6 & {12} & {18} & {24} & {20} & {16} & {12} & 8 & 4 \\ 5 & {10} & {15} & {20} & {25} & {20} & {15} & {10} & 5 \\ 4 & 8 & {12} & {16} & {20} & {24} & {18} & {12} & 6 \\ 3 & 6 & 9 & {12} & {15} & {18} & {21} & {14} & 7 \\ 2 & 4 & 6 & 8 & {10} & {12} & {14} & {16} & 8 \\ 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \end{array}} \right).$$
(8.73)
Note that all the entries of B ij are positive which will prove to be important later on. As we saw for the AE3 case we want to find the possible allowed values for (m1, ⋯, m9), or, equivalently, the possible Dynkin labels[p1, ⋯,p9] for each level .
The corresponding diophantine equation, Equation (8.50), for E10 was found in [47] and reads
$${m^i} = {B^{i3}}\ell - \sum\limits_{j = 1}^9 {{B^{ij}}} {p_j} \geq 0.$$
(8.74)
Since the two sets {p i } and {m i } both consist of non-negative integers and all entries of B ij are positive, these equations put strong constraints on the possible representations that can occur at each level. Moreover, each lowest weight vector Λ() = γ must be a root of E10, so we have the additional requirement
$$({\Lambda ^{(\ell)}}\vert {\Lambda ^{(\ell)}}) = \sum\limits_{i,j = 1}^9 {{B^{ij}}} {p_i}{p_j} - {1 \over {10}}{\ell ^2} \leq 2.$$
(8.75)

The representation content at each level is represented by \(\mathfrak{sl}(10,\, \mathbb{R})\)-tensors whose index structure are encoded in the Dynkin labels [p1, ⋯,p9]. At level = 0 we have the adjoint representation of \(\mathfrak{sl}(10,\, \mathbb{R})\) represented by the generators \({K^a}_b\) obeying the same commutation relations as in Equation (8.32) but now with \(\mathfrak{sl}(10,\, \mathbb{R})\)-indices.

All higher (lower) level representations will then be tensors transforming contravariantly (co-variantly) under the level = 0 generators. The resulting representations are displayed up to level 3 in Table 39. We see that the level 1 and 2 representations have the index structures of a 3-form and a 6-form respectively. In the E10-invariant sigma model, to be constructed in Section 9, these generators will become associated with the time-dependent physical “fields” A abc (t) and \({A_{{a_1}\, \cdots \,{a_6}}}\,(t)\) which are related to the electric and magnetic component of the 3-form in eleven-dimensional supergravity. Similarly, the level 3 generator \({E^{a\vert{b_1}\, \cdots \,{b_9}}}\) with mixed Young symmetry will be associated to the dual of the spatial part of the eleven-dimensional vielbein. This field is therefore sometimes referred to as the “dual graviton”.
Table 39

The low-level representations in a decomposition of the adjoint representation of E10 into representations of its A9 subalgebra obtained by removing the exceptional node in the Dynkin diagram in Figure 49.

Λ() = [p1, ⋯, p9]

Λ() = (m1, ⋯, m10)

A9-representation

E10-generator

1

[0, 0, 1, 0, 0, 0, 0, 0, 0]

(0, 0, 0, 0, 0, 0, 0, 0, 0, 1)

120 1

E abc

2

[0, 0, 0, 0, 0, 1, 0, 0, 0]

(1, 2, 3, 2, 1, 0, 0, 0, 0, 2)

210 2

\({E^{{a_1} \ldots {a_6}}}\)

3

[1, 0, 0, 0, 0, 0, 0, 1, 0]

(1, 3, 5, 4, 3, 2, 1, 0, 0, 3)

440 3

\({E^{a\vert{b_1} \ldots {b_8}}}\)

8.4.1.1 Algebraic structure at low levels
Let us now describe in a little more detail the commutation relations between the low-level generators in the level decomposition of E10 (see Table 39). We recover the Chevalley generators of A9 through the following realisation:
$${e_i} = {K^{i + 1}}_i,\qquad {f_i} = {K^i}_{i + 1},\qquad {h_i} = {K^{i + 1}}_{i + 1} - {K^i}_i\qquad (i = 1, \cdots ,9),$$
(8.76)
where, as before, the \({K^i}_j\)’s obey the commutation relations
$$[{K^i}_j,{K^k}_l] = \delta _j^k{K^i}_l - \delta _l^i{K^k}_j.$$
(8.77)
At levels ±1 we have the positive root generators E abc and their negative counterparts F abc = −τ(E abc ), where τ denotes the Chevalley involution as defined in Section 4. Their transformation properties under the \(\mathfrak{sl}(10,\, \mathbb{R})\)-generators \({K^a}_b\) follow from the index structure and reads explicitly
$$\begin{array}{*{20}c} {[{K^a}_b,{E^{cde}}] = 3\delta _b^{\left[ c \right.}{E^{\left. {de} \right]a}},\quad \quad \quad \quad \quad \quad} \\ {[{K^a}_b,{F_{cde}}] = - 3{\delta ^a}_{\left[ c \right.}{F_{\left. {de} \right]b}},\quad \quad \quad \quad \quad} \\ {[{E^{abc}},{F_{def}}] = 18\delta _{\left[ {de} \right.}^{\left[ {ab} \right.}{K^{\left. c \right]}}_{\left. f \right]} - 2\delta _{def}^{abc}\sum\limits_{a = 1}^{10} {{K^a}_a} ,} \\ \end{array}$$
(8.78)
where we defined
$$\begin{array}{*{20}c} {\delta _{cd}^{ab} = {1 \over 2}(\delta _c^a\delta _d^b - \delta _c^b\delta _d^a)\quad \quad \quad \quad \quad} \\ {\delta _{def}^{abc} = {1 \over {3!}}(\delta _d^a\delta _e^b\delta _f^c \pm 5\;{\rm{permutations}}).} \\ \end{array}$$
(8.79)
The “exceptional” generators e10 and f10 are fixed by Equation (8.76) to have the following realisation:
$${e_{10}} = {E^{123}},\qquad {f_{10}} = {F_{123}}.$$
(8.80)
The corresponding Cartan generator is obtained by requiring [e10, f10] = h10 and upon inspection of the last equation in Equation (8.78) we find
$${h_{10}} = - {1 \over 3}\sum\limits_{i \neq 1,2,3} {K^a}_a + {2 \over 3}({K^1}_1 + {K^2}_2 + {K^3}_3),$$
(8.81)
enlarging \(\mathfrak{sl}(10,\, \mathbb{R})\) to \(\mathfrak{gl}(10,\, \mathbb{R})\).
The bilinear form at level zero is
$$({K^i}_j\vert {K^k}_l) = \delta _l^i\delta _j^k - \delta _j^i\delta _l^k$$
(8.82)
and can be extended level by level to the full algebra by using its invariance, ([x,y]|z) = (x|[y,z]) for x,y,zE10 (see Section 4). For level 1 this yields
$$\left({{E^{abc}}\vert {F_{def}}} \right) = 3!\delta _{def}^{abc},$$
(8.83)
where the normalization was chosen such that
$$\left({{e_{10}}\vert {f_{10}}} \right) = \left({{E^{123}}\vert {F_{123}}} \right) = 1.$$
(8.84)
Now, by using the graded structure of the level decomposition we can infer that the level 2 generators can be obtained by commuting the level 1 generators
$$[{\mathfrak g_1},{\mathfrak g_1}], \subseteq {\mathfrak g_2}.$$
(8.85)
Concretely, this means that the level 2 content should be found from the commutator
$$[{E^{{a_1}{a_2}{a_3}}},{E^{{a_4}{a_5}{a_6}}}].$$
(8.86)
We already know that the only representation at this level is 2102, realized by an antisymmetric 6-form. Since the normalization of this generator is arbitrary we can choose it to have weight one and hence we find
$${E^{{a_1} \cdots {a_6}}} = [{E^{{a_1}{a_2}{a_3}}},{E^{{a_4}{a_5}{a_6}}}].$$
(8.87)
The bilinear form is lifted to level 2 in a similar way as before with the result
$$\left({{E^{{a_1} \cdots {a_6}}}\vert {F_{{b_1} \cdots {b_6}}}} \right) = 6!\delta _{{b_1} \cdots {b_6}}^{{a_1} \cdots {a_6}}.$$
(8.88)
Continuing these arguments, the level 3-generators can be obtained from
$$[[{\mathfrak g_1},{\mathfrak g_1}],{\mathfrak g_1}] \subseteq {\mathfrak g_3}.$$
(8.89)
From the index structure one would expect to find a 9-form generator \({E^{{a_1}\, \cdots \,{a_9}}}\) corresponding to the Dynkin labels [0,0,0,0,0,0,0,0,1]. However, we see from Table 39 that only the representation [1,0,0,0,0,0,0,1,0] appears at level 3. The reason for the disappearance of the representation [0,0,0,0,0,0,0,0,1] is because the generator \({E^{{a_1}\, \cdots \, {a_9}}}\) is not allowed by the Jacobi identity. A detailed explanation for this can be found in [77]. The right hand side of Equation (8.89) therefore only contains the index structure compatible with the generators \({E^{a\vert{b_1}\, \cdots \,{b_8}}}\),
$$[[{E^{a{b_1}{b_2}}},{E^{{b_3}{b_4}{b_5}}}],{E^{{b_6}{b_7}{b_8}}}] = - {E^{[a\vert {b_1}{b_2}]{b_3} \cdots {b_8}}},$$
(8.90)
where the minus sign is purely conventional.
For later reference, we list here some additional commutators that are useful [53]:
$$\begin{array}{*{20}c} {[{E^{{a_1} \cdots {a_6}}},{F_{{b_1}{b_2}{b_3}}}] = - 5!\delta _{{b_1}{b_2}{b_3}}^{\left[ {{a_1}{a_2}{a_3}} \right.}{E^{\left. {{a_1}{a_2}{a_3}} \right]}},\quad \quad \quad \quad \quad \quad \quad} \\ {[{E^{{a_1} \cdots {a_6}}},{F_{{b_1} \cdots {b_6}}}] = 6\cdot6!\delta _{{b_1} \cdots {b_5}}^{\left[ {{a_1} \cdots {a_5}} \right.}{K^{\left. {{a_6}} \right]}}_{\left. {{b_6}} \right]} - {2 \over 3}\cdot6!\delta _{{b_1} \cdots {b_6}}^{{a_1} \cdots {a_6}}\sum\limits_{a = 1}^{10} {{K^a}_a} ,} \\ {[{E^{{a_1}\vert {a_2} \cdots {a_9}}},{F_{{b_1}{b_2}{b_3}}}] = - 7\cdot48\left({\delta _{{b_1}{b_2}{b_3}}^{{a_1}\left[ {{a_2}{a_3}} \right.}{E^{\left. {{a_4} \cdots {a_9}} \right]}} - \delta _{{b_1}{b_2}{b_3}}^{\left[ {{a_2}{a_3}{a_4}} \right.}{E^{\left. {{a_5} \cdots {a_9}} \right]{a_1}}}} \right),} \\ {[{E^{{a_1}\vert {a_2} \cdots {a_9}}},{F_{{b_1} \cdots {b_6}}}] = - 8!\left({\delta _{{b_1} \cdots {b_6}}^{{a_1}\left[ {{a_2} \cdots {a_6}} \right.}{E^{\left. {{a_7}{a_8}{a_9}} \right]}} - \delta _{{b_1} \cdots {b_6}}^{\left[ {{a_2} \cdots {a_7}} \right.}{E^{\left. {{a_8}{a_9}} \right]{a_1}}}} \right).} \\ \end{array}$$
(8.91)

8.4.2 “Gradient representations”

So far, we have only discussed the representations occurring at the first four levels in the E10 decomposition. This is due to the fact that a physical interpretation of higher level fields is yet to be found. There are, however, among the infinite number of representations, a subset of three (infinite) towers of representations with certain appealing properties. These are the “gradient representations”, so named due to their conjectured relation to the emergence of space, through a Taylor-like expansion in spatial gradients [47]. We explain here how these representations arise and we emphasize some of their important properties, leaving a discussion of the physical interpretation to Section 9.

The gradient representations are obtained by searching for “affine representations”, for which the coefficient m9 in front of the overextended simple root of E10 vanishes, i.e., the lowest weights of the representations correspond to the following subset of E10 roots,
$$\gamma = \sum\limits_{i = 1}^8 {{m^i}} {\alpha _i} + \ell {\alpha _{10}}.$$
(8.92)
The Dynkin labels allowed by this restricting are parametrized by an integer k which is related to the level at which a specific representation occurs in the following way:
$$\ell = 3k + 1\qquad [0,0,1,0,0,0,0,0,k],$$
(8.93)
$$\ell = 3k + 2\qquad [0,0,0,0,0,1,0,0,k],$$