Spacelike Singularities and Hidden Symmetries of Gravity
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Abstract
We review the intimate connection between (super)gravity close to a spacelike singularity (the “BKLlimit”) and the theory of Lorentzian KacMoody 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 infinitedimensional KacMoody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a selfcontained 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 infinitedimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian KacMoody algebra. An explicit example is provided for the case of the hyperbolic algebra E_{10}, which is conjectured to be an underlying symmetry of Mtheory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to elevendimensional 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.

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 KacMoody 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 BKLlimit, 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 KacMoody algebra. The dynamical motion in the BKLlimit is then a succession of reflections in the walls bounding the fundamental Weyl chamber and defines “words” in the Weyl group of the KacMoody algebra.
Which billiard region of hyperbolic space actually emerges — and hence which KacMoody 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 elevendimensional supergravity, for which the billiard region is the fundamental region of \({E_{10}} \equiv E_8^{+ +}\), one of the four hyperbolic KacMoody algebras of highest rank 10. The root lattice of E_{10} is furthermore one of the few even, Lorentzian, selfdual lattices — actually the only one in 10 dimensions — a fact that could play a key role in our ultimate understanding of Mtheory.
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 BKLlimit is a robust fact (even though the BKLlimit 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, KacMoody algebras and real forms of Lie algebras.
The appearance of infinite Coxeter groups related to Lorentzian KacMoody 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 selfcontained 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 infinitedimensional KacMoody 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 KacMoody 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 KacMoody algebras in Section 4, and we discuss, in particular, some prominent features of hyperbolic KacMoody algebras.
In Section 5 we then make use of these mathematical concepts to relate the BKL billiards to Lorentzian KacMoody algebras. We show that there is a simple connection between the relevant KacMoody algebra and the Uduality algebras that appear upon toroidal dimensional reduction to three dimensions, when these Uduality algebras are split real forms. The KacMoody algebra is then just the standard overextension of the Uduality algebra in question.
To understand the nonsplit case requires an understanding of real forms of finitedimensional semisimple 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 TitsSatake 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 TitsSatake diagrams. In Section 7 we use these mathematical developments to relate the KacMoody billiards in the nonsplit case to the Uduality algebras appearing in three dimensions.
Up to (and including) Section 7, the developments present wellestablished results. With Section 8 we initiate a journey into more speculative territory. The presence of hyperbolic Weyl groups suggests that the corresponding infinitedimensional KacMoody 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 elevendimensional supergravity. To this end, in Section 8 we introduce the concept of a level decomposition of some of the relevant KacMoody algebras in terms of finite regular subalgebras. This is necessary for studying the sigma model approach to the conjectured infinitedimensional 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 elevendimensional 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 elevendimensional 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 KacMoody 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
The field strength, Equation (2.2), could be modified by additional coupling terms of YangMills or ChaplineManton type [20, 29] (e.g., F_{ C } = dC^{(2)} − C^{(0)}dB^{(2)} for two 2forms C^{(2)} and B^{(2)} and a 0form C^{(0)}, as it occurs in tendimensional type IIB supergravity), but we include these additional contributions to the action in “more”. Similarly, “more” might contain ChernSimons terms, as in the action for elevendimensional supergravity [38].
We shall at this stage consider arbitrary dilaton couplings and menus of pforms. 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 pform menus, spacetime dimensions and dilaton couplings that the billiard region is regular and associated with a KacMoody 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
2.2.1 Action in canonical form
2.2.2 Iwasawa change of variables
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 BKLlimit, which investigates the leading behavior of the fields as x^{0} → ∞ (g → 0). Although the more recent “derivations” of the BKLlimit treat both elements at once [43, 44, 45, 48], it appears useful — especially for rigorous justifications — to separate two aspects of the BKL conjecture^{3}.
The first aspect is that the spatial points decouple in the limit x^{0} → ∞, in the sense that one can replace the Hamiltonian by an effective “ultralocal” Hamiltonian H^{UL} 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 BKLlimit 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
2.3.2 The ultralocal Hamiltonian
2.3.2.1 Comments
 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 elevendimensional 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.
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 BKLlimit 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 BKLlimit 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 motion is thus a succession of futureoriented 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 })^{2} − β^{ i })^{2} + ϕ^{2} = −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 nonchaotic. 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.
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 β^{2} − β^{1} > 0 and β^{3} − β^{2} > 0, then β^{3} − β^{1} > 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 β^{2} > 0 for a 1form with zero dilaton coupling can be written as β^{1} + (β^{2} − β^{1}) > 0 and is thus a consequence of β^{1} > 0 and β^{2} − β^{1} > 0). Finally, one also verifies that in the presence of true pforms (0 < p < d − 1), the gravitational walls are never relevant as they can be written as combinations of pform walls with positive coefficients [49].
 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 socalled 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 offdiagonal components (see also [58]). In the presence of a single pform 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
 1.Gravity brings in the symmetry wallswith i = 1, 2, …,d − 1, and the curvature wall$${\beta ^{i + 1}}  {\beta ^i} = 0,$$(2.48)$$2{\beta ^1} + {\beta ^2} + \cdots + {\beta ^{d  2}} = 0.$$(2.49)
 2.Each pform brings in an electric walland a magnetic wall$${\beta ^1} + \cdots + {\beta ^p} + {{{\lambda ^{(p)}}} \over 2}\phi = 0,$$(2.50)$${\beta ^1} + \cdots + {\beta ^{d  p  1}}  {{{\lambda ^{(p)}}} \over 2}\phi = 0.$$(2.51)
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
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 finitevolume, chaotic case is also called “mixmaster case”, a terminology introduced in four dimensions in [137].

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 fourderivative action in four dimensions, based on R^{2}, is dynamically equivalent to Einstein gravity coupled to a dilaton [160]. Hence, chaos is removed also for this case.

pform gauge fields (0 < p < d − 1) without scalar fields lead to a finitevolume billiard [44].

When both pforms 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 nonzero.
A final comment: How the spatial diffeomorphism constraints and Gauss’ law fit in the conjectured infinitedimensional 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.
 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.
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 (“Kasnerlike 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.”)

In the Kasnerlike 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

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 formwhere 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.$${\pi \over n},$$(3.3)
The group generated by the reflections s_{1}, s_{2} and s_{3} 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)
 One can view PGL(2. ℤ) as the group of fractional transformations of the complex planewith$$C:z \rightarrow z\prime = {{az + b} \over {cz + d}},\qquad a,b,c,d \in {\mathbb Z},$$(3.11)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 halfplane,$$ad  cd = \pm 1.$$(3.12)onto itself whenever the determinant ad − bc of C is equal to +1. This is not the case, however, when det C = −1.$${\rm{{\mathbb H}}} = \{z \in {\rm{{\mathbb C}}}\,\vert \,\Im z > 0\} ,$$(3.13)
 For this reason, it is convenient to consider alternatively the following action of PGL(2. ℤ),(a. b. c. d ∈ ℤ), which does map the complex upperhalf plane onto itself, i.e., which is such that \(\mathfrak I {z\prime} > 0\) whenever \(\mathfrak I z > 0\).$$\begin{array}{*{20}c} {z \rightarrow z\prime = {{az + b} \over {cz + d}},\quad \,\;\;\;\;\;\;{\rm{if}}\;ad  cb = 1,} \\ {{\rm{or}}} \\ {z \rightarrow z\prime = {{a\bar z + b} \over {c\bar z + d}},\quad \quad \,{\rm{if}}\;ad  cb =  1,} \\\end{array}$$(3.14)
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).
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) KacMoody algebra \(A_1^{+ +}\). Our first encounter with Lorentzian KacMoody algebras in more general gravitational theories will also be through their Weyl groups, which are, exactly as in the fourdimensional case just described, particular instances of (nonEuclidean) 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 KacMoody 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 I_{2}(3) = A_{2}
Consider the dihedral group I_{2}(3) of order 6 of symmetries of the equilateral triangle in the Euclidean plane.
The dihedral group I_{2}(3) is also denoted A_{2} 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 \({I_2}(\infty) \equiv A_1^ +\)
3.2.2 Definition
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 (i ≠ j), the exponent m_{ ij } is equal to 2. We have drawn the Coxeter graphs for the Coxeter groups I_{2}(3), I_{2}(m) and for the Coxeter group H_{3} 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 xaxis (s_{1}), (ii) the straight line joining the points (1,0) and (0,1) (s_{2}), and (iii) the yaxis (s_{3}). The Coxeter exponents are finite and equal to 4 (m_{12} = m_{21} = m_{23} = m_{32} = 4) and 2 (m_{13} = m_{31} = 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 s_{2}s_{1}s_{2} is a reflection in the line parallel to the yaxis going through (1, 0) and thus the product t = s_{2} s_{1}s_{2}s_{3} is a translation by +2 in the xdirection. 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
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 I_{2}(3), the identity has length zero, the generators s_{1} and s_{2} have length one, the two nontrivial rotations have length two, and the third reflection s_{3} has length three. Note that the rotations have representations involving two and four (and even a higher number of) generators since for instance s_{1}s_{2} = s_{2}s_{1}s_{2}s_{1}, 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 s_{3} = s_{1}s_{2}s_{1} = s_{2}s_{1} s_{2}. Both involve three generators and define the length of s_{3} 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.
 1.
The induced scalar product on Π is nondegenerate and in fact positive definite, or
 2.
the induced scalar product is positive semidefinite, i.e., there is a null direction orthogonal to any other direction.

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 } + 2kλ ≠ α_{ i } unless k = 0.
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 nontrivial direct sum, i.e., if one can slit the indices i, j in two sets J and Λ such that A_{ ij } = 0 whenever i ∈ J, j ∈ Λ or i ∈ Λ, j ∈ J. 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
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
3.3 Finite Coxeter groups
Finite Coxeter groups.
Name  Coxeter graph 

A _{ n }  
B_{ n } ≡ C_{ n }  
D _{ n }  
I _{2(m)}  
F _{2}  
E _{6}  
E _{7}  
E _{8}  
H _{3}  
H _{4} 
3.4 Affine Coxeter groups
Affine Coxeter groups.
Name  Coxeter graph 

\(A_1^ +\)  
\(A_n^ + \,(n > 1)\)  
\(B_n^ + \,(n > 2)\)  
\(C_n^ +\)  
\(D_n^ +\)  
\(G_2^ +\)  
\(F_4^ +\)  
\(E_6^ +\)  
\(E_7^ +\)  
\(E_8^ +\) 
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.
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 x^{0} = 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).
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(ω_{i}, 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 (nonhyperbolic) Lorentzian algebras.
The following theorem enables one to decide whether a Coxeter group is hyperbolic by mere inspection of its Coxeter graph.

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 nonirreducible 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.)

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 j ≠ i form a basis of the hyperplane Π_{ i } orthogonal to ω_{ i }. Because ω_{ i } is nonspacelike (the group is hyperbolic), the hyperplane Π_{ i } is spacelike or null. The Coxeter graph defined by the α_{ j } with j ≠ i (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 nonempty. Let Π_{ i } be the hyperplane spanned by the α_{ j } with j ≠ i, 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 twodimensional 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 nontimelike. 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 nonspacelike). This concludes the proof of the theorem.
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.
Hyperbolic Coxeter groups of rank 4.
Hyperbolic Coxeter groups of rank 5.
Hyperbolic Coxeter groups of rank 6.
Hyperbolic Coxeter groups of rank 7.
Hyperbolic Coxeter groups of rank 8.
Hyperbolic Coxeter groups of rank 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 futurepointing nonspacelike 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(α_{ 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.

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 fourdimensional 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].

Nonhyperbolic Lorentzian Coxeter groups are associated through the above construction with infinitevolume 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 KacMoody 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.

\({m_{ij}} = 3 \Rightarrow {c_i} = {c_j}\).

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

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

\({m_{ij}} = \infty \Rightarrow {c_i} = {c_j}\).
The rules are consistent as can be seen by starting from an arbitrary node, say α_{1}, for which one takes c_{1} = 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 α_{1} 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 E_{10} and DE_{10} are unique up to an overall rescaling because the nontrivial m_{ ij } (i ≠ j) 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 BE_{10} 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 KacMoody 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 KacMoody algebras.
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 KacMoody algebras (defined in Section 4.1). Indeed, there is an intimate connection between crystallographic Coxeter groups and KacMoody algebras. This connection might be familiar in the finite case. For instance, it is well known that the finite symmetry group A_{2} of the equilateral triangle (isomorphic to the group of permutations of 3 objects) and the corresponding hexagonal pattern of roots are related to the finitedimensional Lie algebra \(\mathfrak {sl}(3,\,\mathbb R)\) (or \(\mathfrak {su}\)(3)). The group A_{2} 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 crystallographic Coxeter group is the Weyl group of a KacMoody 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 KacMoody algebras and to explain the connection between Coxeter groups and KacMoody algebras. How this is relevant to gravitational theories will be discussed in Section 5.
4.1 Definitions
 1.
For each i = 1, …, n, one associates a node in the diagram.
 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.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:
 (a)
one draws a number of lines between i and j equal to max( A_{ ij },  A_{ ij });
 (b)
one draws an arrow from j to i if A_{ ij }  > A_{ ij }.
 (a)
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 KacMoody algebra when det A = 0 is then slightly more involved.
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.
Finite Lie algebras.
Name  Dynkin diagram 

A _{ n }  
B _{ n }  
C _{ n }  
D _{ n }  
G _{2}  
F _{4}  
E _{6}  
E _{7}  
E _{8} 
When the Cartan matrix A is of Lorentzian signature the KacMoody algebra \({\mathfrak {g}(A)}\), constructed from A using the ChevalleySerre relations, is called a Lorentzian KacMoody algebra. This is the case of main interest for the remainder of this paper.
4.2 Roots
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
Finally, we quote the following useful theorem.
Theorem: The KacMoody 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 nontrivial ideals^{10} (see [116] and Section 4.5).
4.4 Three examples
 A_{2}: We start with A_{2}, the Cartan matrix of which is Equation (4.4). The defining relations are then:The commutator [e_{1}, e_{2}] 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 A_{2} is eightdimensional and one may take as basis {h_{1}, h_{2}, e_{1}, e_{2}, [e_{1}, e_{2}], f_{1}, f_{2}, [f_{1}, f_{2}]}. The vector [e_{1}, e_{2}] corresponds to the positive root α_{1} + α_{2}.$$\begin{array}{*{20}c} {[{h_1},{h_2}] = 0,\quad \;\;} & {[{h_1},{e_1}] = 2{e_1},\;\;} & {[{h_1},{e_2}] =  {e_2},\quad} \\ {[{h_1},{f_1}] =  2{f_1},} & {[{h_1},{f_2}] = {f_2},\quad} & {[{h_2},{e_1}] =  {e_1},\quad} \\ {[{h_2},{e_2}] = 2{e_2},\;\;} & {[{h_2},{f_1}] = {f_1},\quad} & {[{h_2},{f_2}] =  2{f_2},\;\;} \\ {[{e_1},[{e_1},{e_2}]] = 0,\;\;\quad \quad \;\;} & {[{e_2},[{e_2},{e_1}]] = 0,\quad \quad \quad} & {[{f_1},[{f_1},{f_2}]] = 0,\quad \quad \quad \quad} \\ {[{f_2},[{f_2},{f_1}]] = 0\;\;\quad \quad \quad \;} & {[{e_i},{f_j}] = {\delta _{ij}}{h_j}.} & {} \\ \end{array}$$(4.20)
 B_{2}: The algebra B_{2}, the Cartan matrix of which is Equation (4.5), is defined by the same set of generators, but the Serre relations are now [e_{1}, [e_{1}, [e_{1}, e_{2}]]] = 0 and [e_{2}, [e_{2}, e_{1}]] = 0 (and similar relations for the f’s). The algebra is still finitedimensional and contains, besides the generators, the commutators [e_{1}, e_{2}], [e_{1}, [e_{1}, e_{2}]], their negative counterparts [f_{1}, f_{2}] and [f_{1}, [f_{1}, f_{2}]], and nothing else. The triple commutator [e_{1}, [e_{1}, [e_{1}, e_{2}]]] vanishes by the Serre relations. The other triple commutator [e_{2}, [e_{1}, [e_{1}, e_{2}]]] vanishes also by the Jacobi identity and the Serre relations,(Each term on the righthand side is zero: The first by antisymmetry of the bracket and the second because [e_{2}, [e_{1}, e_{2}]] = −[e_{2}, [e_{2}, e_{1}]] = 0.) The algebra is 10dimensional and is isomorphic to \({\mathfrak {so}(3,\,2)}\).$$[{e_1},[{e_1},[{e_1},{e_2}]]] = [[{e_2},{e_1}],[{e_1},{e_2}]] + [{e_1},[{e_2},[{e_1},{e_2}]]] = 0.$$
 \(A_1^{+}\): 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 A_{2}, but with Serre relations given by(and similar relations for the f’s). This innocentlooking change in the Serre relations has dramatic consequences because the corresponding algebra is infinitedimensional. (We analyze here the algebra generated by the h’s, e’s and f’s, which is in fact the derived KacMoody algebra — see Section 4.5 on affine KacMoody algebras. The derived algebra is already infinitedimensional.) To see this, consider the \({\mathfrak {sl}(2,\,\mathbb R)}\) current algebra, defined 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)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 \({\mathfrak {sl}(2,\,\mathbb R)}\),$$[J_m^a,J_n^b] = {f^{ab}}_cJ_{m + n}^c + m{k^{ab}}c{\delta _{m + n,0}},$$(4.22)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 multicommutator involving them. The map$$[J_0^3,J_0^ + ] = 2J_0^ + ,\qquad [J_0^3,J_0^  ] =  2J_0^  ,\qquad [J_0^ + ,J_0^  ] = J_0^3.$$preserves the defining relations of the KacMoody algebra and defines an isomorphism of the (derived) KacMoody algebra with the current algebra. The KacMoody algebra is therefore infinitedimensional. One can construct nonvanishing infinite multicommutators, in which e_{1} and e_{2} alternate:$$\begin{array}{*{20}c} {{h_1}} & \rightarrow & {J_0^3,} & {{h_2}} & \rightarrow & { J_0^3 + c,} \\ {{e_1}} & \rightarrow & {J_0^ + ,} & {{e_2}} & \rightarrow & {J_1^  ,} \\ {{f_1}} & \rightarrow & {J_0^  ,} & {{f_2}} & \rightarrow & {J_{ 1}^ +} \\ \end{array}$$(4.23)The Serre relations do not cut the chains of multicommutators to a finite number.$$\begin{array}{*{20}c} {\left[ {{e_1},} \right.\left[ {{e_2},} \right.\left[ {{e_1}, \cdots ,\left[ {{e_1},{e_2}} \right]\left. {\left. \cdots \right]} \right]} \right]\sim J_n^3\;\;\;} & {(n\;{e_1}\prime{\rm{s}}\;{\rm{and}}\;n\;{e_2}\prime{\rm{s}}),\;\;\;\;} \\ {\left[ {{e_1},} \right.\left[ {{e_2},} \right.\left[ {{e_1}, \cdots ,\left[ {{e_2},{e_1}} \right]\left. {\left. \cdots \right]} \right]} \right]\sim J_n^ + \;\;} & {(n + 1\;{e_1}\prime{\rm{s}}\;{\rm{and}}\;n\;{e_2}\prime{\rm{s}}),\;} \\ {\left[ {{e_2},} \right.\left[ {{e_1},} \right.\left[ {{e_2}, \cdots ,\left[ {{e_1},{e_2}} \right]\left. {\left. \cdots \right]} \right]} \right]\sim J_{n + 1}^ } & {(n\;{e_1}\prime{\rm{s}}\;{\rm{and}}\;n + 1\;{e_2}\prime{\rm{s}}).} \\ \end{array}$$(4.24)
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 semidefinite (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 onedimensional and that the ranks of S and A are equal to n − 1.
Untwisted affine KacMoody algebras.
Name  Dynkin diagram 

\(A_1^ +\)  
\(A_n^ + \,(n > 1)\)  
\(B_n^ +\)  
\(C_n^ +\)  
\(D_n^ +\)  
\(G_2^ +\)  
\(F_4^ +\)  
\(E_6^ +\)  
\(E_7^ +\)  
\(E_8^ +\) 
Twisted affine KacMoody algebras. We use the notation of Kac [116].
Name  Dynkin diagram 

\(A_2^{(2)}\)  
\(A_{2n}^{(2)}\,(n \geq 2)\)  
\(A_{2n  1}^{(2)}\,(n \geq 3)\)  
\(D_{n + 1}^{(2)}\)  
\(E_6^{(2)}\)  
\(D_4^{(3)}\) 
4.6 The invariant bilinear form
4.6.1 Definition
4.6.2 Real and imaginary roots
When the algebra is infinitedimensional, the invariant scalar product does not have Euclidean signature. The spacelike roots are called “real roots”, the nonspacelike ones are called “imaginary roots” [116]. While the real roots are nondegenerate (i.e., the corresponding eigenspaces, called “root spaces”, are onedimensional), this is not so for imaginary roots. In fact, it is a challenge to understand the degeneracy of imaginary roots for general indefinite KacMoody algebras, and, in particular, for Lorentzian KacMoody algebras.
Another characteristic feature of real roots, familiar from standard finitedimensional 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 nontrivial 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 finitedimensional case, this is no longer true even for real roots in the case of infinitedimensional KacMoody algebras^{11}. When all the real roots have the same length, one says that the algebra is “simplylaced”. 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
4.6.4 The generalized Casimir operator
From the invariant bilinear form, one can construct a generalized Casimir operator as follows.
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 generators^{12}.
4.6.4.1 Note
4.7 The Weyl group

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 (i ≠ j).
Cartan integers and Coxeter exponents.
A _{ ij } A _{ ji }  m _{ ij } 

0  2 
1  3 
2  4 
3  6 
≥  ∞ 
Because the Weyl groups are (crystallographic) Coxeter groups, we can use the theory of Coxeter groups to analyze them. In the KacMoody context, the fundamental region is called “the fundamental Weyl chamber”.
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 KacMoody algebras do not have immediate analogs on the Coxeter side.
4.7.1 Examples
 Consider the Cartan matricesAs 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 {α′_{1}, α′_{2}, α′_{3}} and {α″_{1}, α″_{2}, α″_{3}}, respectively. In both cases, the Coxeter exponents are m_{12} = ∞, m_{13} = 2, m_{23} = 3 and the metric B_{ ij } of the geometric Coxeter construction is$$A\prime = \left({\begin{array}{*{20}c} 2 & { 2} & 0 \\ { 2} & 2 & { 1} \\ 0 & { 1} & 2 \\ \end{array}} \right),\qquad A\prime \prime = \left({\begin{array}{*{20}c} 2 & { 4} & 0 \\ { 1} & 2 & { 1} \\ 0 & { 1} & 2 \\ \end{array}} \right)$$We associate the simple roots {α_{1}, α_{2}, α_{3}} 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 KacMoody algebras described by the Cartan matrices A′ and A″.$$A\prime = \left({\begin{array}{*{20}c} 1 & { 1} & 0 \\ { 1} & 1 & { {1 \over 2}} \\ 0 & { {1 \over 2}} & 1 \\ \end{array}} \right).$$Choosing the longest KacMoody roots to have squared length equal to two yields the scalar productsRecall now from Section 3 that the fundamental reflections \({\sigma _i} \in \mathfrak{B}\) have the following geometric realisation$$S\prime = \left({\begin{array}{*{20}c} 2 & { 2} & 0 \\ { 2} & 2 & { 1} \\ 0 & { 1} & 2 \\ \end{array}} \right),\qquad S\prime \prime = \left({\begin{array}{*{20}c} {{1 \over 2}} & { 1} & 0 \\ { 1} & 2 & { 1} \\ 0 & { 1} & 2 \\ \end{array}} \right).$$which in this case becomes$${\sigma _i}({\alpha _j}) = {\alpha _j}  2{B_{ij}}{\alpha _i}\qquad (i = 1,2,3),$$(4.60)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 }$$\begin{array}{*{20}c} {{\sigma _1}:} & {{\alpha _1} \rightarrow  {\alpha _1},\quad \quad} & {{\alpha _2} \rightarrow {\alpha _2} + 2{\alpha _1},\;} & {{\alpha _3} \rightarrow {\alpha _3},\quad \quad} \\ {{\sigma _2}:} & {{\alpha _1} \rightarrow {\alpha _1} + 2{\alpha _2},} & {{\alpha _2} \rightarrow  {\alpha _2},\quad \quad} & {{\alpha _3} \rightarrow {\alpha _3} + {\alpha _2},} \\ {{\sigma _3}:} & {{\alpha _1} \rightarrow {\alpha _1},\quad \quad} & {{\alpha _2} \rightarrow {\alpha _2} + {\alpha _3},\;} & {{\alpha _3} \rightarrow  {\alpha _3}.\quad \;} \\\end{array}$$while the Weyl group \(\mathfrak{W}[A_2^{(2) +}]\) acts as$$\begin{array}{*{20}c} {{{r\prime}_1}:} & {{{\alpha \prime}_1} \rightarrow  {{\alpha \prime}_1},\quad \quad} & {{{\alpha \prime}_2} \rightarrow {{\alpha \prime}_2} + 2{{\alpha \prime}_1},\;} & {{{\alpha \prime}_3} \rightarrow {{\alpha \prime}_3},\quad \quad} \\ {{{r\prime}_2}:} & {{{\alpha \prime}_1} \rightarrow {{\alpha \prime}_1} + 2{{\alpha \prime}_2},} & {{{\alpha \prime}_2} \rightarrow  {{\alpha \prime}_2},\quad \quad} & {{{\alpha \prime}_3} \rightarrow {{\alpha \prime}_3} + {{\alpha \prime}_2},} \\ {{{r\prime}_3}:} & {{{\alpha \prime}_1} \rightarrow {{\alpha \prime}_1},\quad \quad} & {{{\alpha \prime}_2} \rightarrow {{\alpha \prime}_2} + {{\alpha \prime}_3},\quad} & {{{\alpha \prime}_3} \rightarrow  {{\alpha \prime}_3},\quad} \\ \end{array}$$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$$\begin{array}{*{20}c} {r\prime {\prime _1}:} & {\alpha \prime {\prime _1} \rightarrow  \alpha \prime {\prime _1},\;\;\quad} & {\alpha \prime {\prime _2} \rightarrow \alpha \prime {\prime _2} + 4\alpha \prime {\prime _1},\;} & {\alpha \prime {\prime _3} \rightarrow \alpha \prime {\prime _3},\;\;\quad \quad \;} \\ {r\prime {\prime _2}:} & {\alpha \prime {\prime _1} \rightarrow \alpha \prime {\prime _1} + \alpha \prime {\prime _2},} & {\alpha \prime {\prime _2} \rightarrow  \alpha \prime {\prime _2},\;\;\quad \quad} & {\alpha \prime {\prime _3} \rightarrow \alpha \prime {\prime _3} + \alpha \prime {\prime _2},\;} \\ {r\prime {\prime _3}:} & {\alpha \prime {\prime _1} \rightarrow \alpha \prime {\prime _1},\;\;\quad \;\;} & {\alpha \prime {\prime _2} \rightarrow \alpha \prime {\prime _2} + \alpha \prime {\prime _3},\;\;} & {\alpha \prime {\prime _3} \rightarrow  \alpha \prime {\prime _3}.\;\;\;\;\;\;\;} \\ \end{array}$$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.$$Q\prime = \sum\limits_i {{\mathbb Z}\alpha \prime _i}\qquad {\rm{and}}\qquad Q\prime \prime = \sum\limits_i {{\mathbb Z}\alpha \prime \prime _i},$$(4.61)It follows, of course, that the Weyl groups of the KacMoody 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 matrixand its symmetrization$$A\prime \prime \prime = \left({\begin{array}{*{20}c} 2 & { 6} & 0 \\ { 1} & 2 & { 1} \\ 0 & { 1} & 2 \\\end{array}} \right),$$The Weyl group \(\mathfrak{W}[A\prime \prime \prime ]\) of the corresponding KacMoody 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$$S\prime \prime \prime = \left({\begin{array}{*{20}c} {{1 \over 3}} & { 1} & 0 \\ { 1} & 2 & { 1} \\ 0 & { 1} & 2 \\ \end{array}} \right),$$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.$$\begin{array}{*{20}c} {r\prime \prime {\prime _1}}: & {\alpha \prime \prime {\prime _1} \rightarrow  \alpha \prime \prime {\prime _1},\;\;\quad \;} & {\alpha \prime \prime {\prime _2} \rightarrow \alpha \prime \prime {\prime _2} + 6\alpha \prime \prime {\prime _1},} & {\alpha \prime \prime {\prime _3} \rightarrow \alpha \prime \prime {\prime _3}\quad \quad \quad} \\ {r\prime \prime {\prime _2}}: & {\alpha \prime \prime {\prime _1} \rightarrow \alpha \prime \prime {\prime _1} + \alpha \prime \prime {\prime _2},} & {\alpha \prime \prime {\prime _2} \rightarrow  \alpha \prime \prime {\prime _2}\quad \;\;\quad} & {\alpha \prime \prime {\prime _3} \rightarrow \alpha \prime \prime {\prime _3} + \alpha \prime \prime {\prime _2}\;} \\ {r\prime \prime {\prime _3}}:& {\alpha \prime \prime {\prime _1} \rightarrow \alpha \prime \prime {\prime _1},\;\quad \quad} & {\alpha \prime \prime {\prime _2} \rightarrow \alpha \prime \prime {\prime _2} + \alpha \prime \prime {\prime _3},} & {\alpha \prime \prime {\prime _3} \rightarrow  \alpha \prime \prime {\prime _3}\quad \quad} \\ \end{array}$$
4.8 Hyperbolic KacMoody algebras
Hyperbolic KacMoody algebras are by definition Lorentzian KacMoody 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 KacMoody 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 KacMoody 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 KacMoody algebra that has this given Coxeter group as Weyl group.
The hyperbolic KacMoody 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 \(C{E_{10}} \equiv A_{15}^{(2) +}\), BE_{10} and CE_{10} 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 KacMoody 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 }Λ_{ j }) ≥ 0 (with, for any K, strict inequality for at least one i) and hence, since they are nonspacelike, must lie in the future light cone. Recall indeed that the scalar product of two nonspacelike vectors with the same time orientation is nonpositive. 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 Weylconjugated 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 KacMoody 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 KacMoody algebra such that (α_{ i }α_{ i }) = 2 for all simple roots α_{ i }. Then, any point α on the root lattice Q with (αα) ≤ 2 is a root (note that (αα) is even). In particular, the set of real roots is the set of points on the root lattice with (αα) = 2, while the set of imaginary roots is the set of points on the root lattice (minus the origin) with (αα) ≤ 0. For a proof, see [116], Chapter 5.
The version of this theorem applicable to KacMoody algebras with different simple root lengths is the following.

The set of all short real roots is {α ∈ Q (αα) = 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 (αα) ≤ 0.
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
4.8.3.1 The KacMoody Algebra \(A_1^{+ +}\)
Applying the first theorem, one easily verifies that the only positive roots at level zero are the roots kα_{1} + mα_{2}, k − m ≤ 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 k − m = 1, the root is real and has length squared equal to two.
Similarly, the only roots at level one are (m + a)α_{1} + mα_{2} + α_{3} with a^{2} ≤ m, 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)α_{1} + mα_{2} + α_{3} with a^{2} < m are imaginary and have squared length equal to 2(a^{2} + 1 − m) ≤ 0. In particular, the root m(α_{1} + α_{2}) + α_{3} 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 (βα_{i}) ≤ 0). When m = 1, none of the level1 roots is in \(\mathcal{F}\) and is either in the Weyl orbit of α_{1} + α_{2}, or in the Weyl orbit of α_{3}.
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)α_{1} + mα_{2} + 2α_{3} and m(α_{1} + α_{2}) + 2α_{3} 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 KacMoody Algebra \(A_2^{(2) +}\)
The real roots, which are Weyl conjugate to one of the simple roots α_{1} or α_{2} (α_{3} is in the same Weyl orbit as α_{2}), 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 α_{1} should be a multiple of 4. The short real roots are the vectors on the root lattice with length squared equal to onehalf. The imaginary roots are all the vectors on the root lattice with length squared ≤ 0.
We define again the level ℓ as counting the number of times the root α_{3} occurs. The positive roots at level zero are the positive roots of the twisted affine algebra \(A_2^{(2)}\), namely, α_{1} and (2m + a)α_{1}+mα_{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)α_{1} + mα_{2} are not long real roots when m is even because they fail to satisfy the condition that the coefficient (2m ± 2) of α_{1} is a multiple of 4. The roots at level zero are all real, except when a = 0, in which case the roots m(2α_{1} + α_{2}) have zero norm.
To get the long real roots at level one, we first determine the vectors α = α_{3} + kα_{1} + mα_{2} of squared length equal to two. The condition (αα) = 2 easily leads to m = p^{2} for some integer p ≥ 0 and k = 2p^{2} ± 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 (2p^{2} + 1)α_{1} + (p^{2} + p + 1)α_{2} + α_{3} and (2p^{2} + 4p + 3)α_{1} + (p^{2} + p + 1)α_{2} + α_{3} for p a nonnegative integer.
Finally, the imaginary roots at level one in the fundamental domain \(\mathcal{F}\) read (2m − 1)α_{1} + mα_{2} + α_{3} and 2mα_{1} + mα_{2} + α_{3} 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 finitedimensional Lie algebras
An interesting class of Lorentzian KacMoody algebras can be constructed by adding simple roots to finitedimensional 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, finitedimensional, simple Lie algebra of rank r, with simple roots α_{1}, ⋯, 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 G_{2})). The roots of simplylaced algebras are regarded as long roots.
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 KacMoody algebra.
The root α_{0} is called the affine root and the algebra \(\mathfrak{g}^+\) (\(\mathfrak{g}^{(1)}\) in Kac’s notations [116]) with roots α_{0}, α_{1}, ⋯, 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 α_{0}. One has explicitly (α_{−1} α_{−1}) = 2 = (α_{0}α_{0}) and (α_{−1} α_{0}) = −1, which shows that the overextended root is attached to the affine root (and only to the affine root) with a single link.

\(A_k^{+ +}(k \leq 7)\),

\(B_k^{+ +}(k \leq 8)\),

\(C_k^{+ +}(k \leq 4)\),

\(D_k^{+ +}(k \leq 8)\),

\(G_2^{+ +}\),

\(F_4^{+ +}\),

\(E_k^{+ +}(k = 6,\,7,\,8)\).
4.9.1.1 A special property of E_{10}
Of these maximal rank hyperbolic algebras, E_{10} 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 E_{10} is selfdual, i.e., that the fundamental weights belong to the root lattice of E_{10}. In view of the above theorem on roots of hyperbolic algebras and of the hyperboliticity of E_{10}, the fundamental weights of E_{10} are actually (imaginary) roots since they are nonspacelike. The root lattice of E_{10} is the only Lorentzian, even, selfdual 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”.
 1.
If α ∈ Δ, then −α ∈ Δ.
 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.
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.
If α, β ∈ Δ are not proportional to each other and (αα) ≤ (ββ) then A_{ α,β } = 0, ±1. Moreover if (αβ) = 0, then (ββ) = (αα), 2 (αα), or 3 (αα).
 5.
If If α, β ∈ Δ, but α — β ∉ Δ ∪ 0, then (αβ) ≤ 0 and, as a consequence, if α, β ∈ Δ but α ± β ∉ Δ ∪ 0 then (αβ) = 0. That (αβ) ≤ 0 can be seen as follows. Clearly, α and β can be assumed to be linearly independent^{14}. Now, assume (αβ) > 0. By the previous point, A_{ α,β } = 1 or A_{ β,α } = 1. But then either α − A_{ α,β }β = α − β ∈ Δ or −(β − A_{ β,α }α) = α − β ∈ Δ by (4.72), contrary to the assumption. This proves that (αβ) ≤ 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].

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 finitedimensional simple Lie algebras (A_{ n } with n ≥ 1, B_{ n } with n ≥ 3, C_{ n } with n ≥ 2, D_{ n } with n ≥ 4, G_{2}, F_{4}, E_{6}, E_{7} and E_{8}).

Irreducible nonreduced root systems are all given by the socalled (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 ndimensional 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 k ≠ k′ 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 α_{1}, to indicate that 2α_{1} is also a root.
It is sometimes convenient to rescale the roots by the factor \((1/\sqrt 2)\) so that the highest root θ = 2(α_{1} + α_{2} + ⋯ + 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 nonvanishing 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 onehalf the highest root of C_{ r } to make the extension (see [116], formula in Paragraph 6.4, bottom of page 84). The formulas for α_{0} and α_{−1} are 2α_{0} = u−θ and 2α_{−1} = −u−v (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 threedimensional gravity a coset model \({\mathcal G}/{\mathcal K}({\mathcal G})\), where the socalled 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

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 CE_{10}. It is the fourth rank10 hyperbolic algebra, besides E_{10}, BE_{10} and DE_{10}.

The overextensions \(E_6^{(2) +}\) (rank 6) and \(D_4^{(3) +}\) (rank 4) are hyperbolic.
Twisted overextended KacMoody algebras.
Name  Dynkin diagram 

\(A_2^{(2) +}\)  
\(A_2^{(2)\prime}\)  
\(A_{2n}^{(2) +}\,(n \geq 2)\)  
\(A_{2n}^{(2)\prime +}\,(n \geq 2)\)  
\(A_{2n  1}^{(2) +}\,(n \geq 3)\)  
\(D_{n + 1}^{(2) +}\)  
\(E_6^{(2) +}\)  
\(D_4^{(3) +}\) 
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 KacMoody algebras are symmetrizable, the ones that are obtained through the process of overextension are.
4.9.4 Algebras of GaberdielOliveWest type
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 KacMoody algebras
This section is based on [96].
4.10.1 Definitions
Let \({\mathfrak g}\) be a KacMoody 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 \({\mathfrak g}\).
4.10.2 Examples — Regular subalgebras of E_{10}
4.10.2.1 \({A_9} \subset {\mathcal B} \subset{E_{10}}\)
In terms of the simple roots of E_{10}, the simple roots of \({\mathcal B}\) are α_{1} through α_{9} and ᾱ_{10} = 2α_{10} + α_{1} + 2α_{2} + 3α_{3} + 2α_{4} + α_{5}. 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 DE_{10} ⊂ E_{10}
In [67], Dynkin has given a method for finding all maximal regular subalgebras of finitedimensional simple Lie algebras. The method is based on using the highest root and is not generalizable as such to general KacMoody algebras for which there is no highest root. Nevertherless, it is useful for constructing regular embeddings of overextensions of finitedimensional simple Lie algebras. We illustrate this point in the case of E_{8} and its overextension \({E_{10}} \equiv E_8^{+ +}\). In the notation of Figure 21, the simple roots of E_{8} (which is regularly embedded in E_{10}) are α_{1}, ⋯, α_{7} and α_{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 elevendimensional supergravity on a circle, one gets, after dropping the KaluzaKlein vector and the 3form, the bosonic sector of pure \({\mathcal N} = 1\) tendimensional supergravity. The simple roots of DE_{10} are the symmetry walls and the electric and magnetic walls of the 2form and coincide with the positive roots given above [45]. A similar construction shows that \(A_8^{+ +}\) can be regularly embedded in E_{10}, and that DE_{10} can be regularly embedded in \(B_8^{+ +}\). See [106] for a recent discussion of DE_{10} 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 finitedimensional 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 KacMoody algebras in [76]. It covers also nonmaximal regular subalgebras and goes as follows:
Theorem: Let \(\Phi _{{\rm{real}}}^ +\) be the set of positive real roots of a KacMoody 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 nonvanishing determinant. For each 1 ≤ i ≤ n, choose nonzero root vectors E_{ i } and F_{ i } in the onedimensional 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 KacMoody 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 nonpositive (because γ_{ i } − γ_{ j } is not a root), so that [C_{ ij }] is a Cartan matrix.
4.10.3.1 Comments
 1.
When the Cartan matrix is degenerate, the corresponding KacMoody algebra has nontrivial ideals [116]. Verifying that the ChevalleySerre relations are fulfilled is not sufficient to guarantee that one gets the KacMoody algebra corresponding to the Cartan matrix [C_{ ij }] since there might be nontrivial quotients. Situations in which the algebra generated by the set {E_{ i }, F_{ i }, H_{ i }} is the quotient of the KacMoody algebra with Cartan matrix [C_{ ij }] by a nontrivial ideal were discussed in [96].
 2.
If the matrix [C_{ ij }] is decomposable, say C = D ⊕ E with D and E indecomposable, then the KacMoody algebra \({\mathbb K}{\mathbb M}(C)\) generated by C is the direct sum of the KacMoody algebra \({\mathbb K}{\mathbb M}(D)\) generated by D and the KacMoody 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 nonvanishing determinant, then both D and E have nonvanishing 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.
5 KacMoody 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 BKLlimit, 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 twodimensional 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 threedimensional gravity coupled to a nonlinear sigmamodel 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 Uduality group” [142]. This situation covers the cases of pure gravity in any spacetime dimension, as well as all known supergravity 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 KacMoody algebra. The billiard table is then the fundamental Weyl chamber of a Lorentzian KacMoody algebra [45, 46] and the billiard is also called a “KacMoody billiard”. This enables one to reformulate the dynamics as a motion in the Cartan subalgebra of the Lorentzian KacMoody algebra, hinting at the potential — and still conjectural at this stage — existence of a deeper, infinitedimensional symmetry of the theory.
 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.
Second, we show how to reformulate the dynamics as a motion in the Cartan subalgebra of a Lorentzian KacMoody algebra.
 3.
Finally, we connect the Lorentzian KacMoody algebra that appears in the BKLlimit 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
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 KacMoody billiards since the Coxeter groups are the Weyl groups of the KacMoody algebras \(A_{d  2}^{+ +}\).
5.1.2 The Coxeter billiard for the coupled gravity3Form system Coxeter polyhedra
Let us review the conditions that must be fulfilled in order to get a KacMoody 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 \({{\mathcal H}_{d1}}\). In the general case, the dihedral angles between adjacent faces of \({{\mathcal H}_{d+n1}}\) can take arbitrary continuous values, which depend on the dilaton couplings, the spacetime dimensions and the ranks of the pforms 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 \({{\mathcal H}_{d+n1}}\) 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 \({{\mathcal H}_{d+n1}}\).
In order for the billiard region to be identifiable with the fundamental Weyl chamber of a KacMoody algebra, the Coxeter polyhedron should be a simplex, i.e., bounded by d + n walls in a d + n − 1dimensional space. In general, the Coxeter polyhedron need not be a simplex.
These conditions are very restrictive and hence gravitational theories which can be mapped to a KacMoody algebra in the BKLlimit are rare.
5.1.2.1 The Coxeter billiard of elevendimensional supergravity
Consider for instance the action (2.1) for gravity coupled to a single threeform in D = d + 1 spacetime dimensions. We assume D ≥ 6 since in lower dimensions the 3form is equivalent to a scalar (D = 5) or has no degree of freedom (D < 5).
Theorem: Whenever a pform (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 pforms. (These walls are all listed in Section 2.5.)

the magnetic wall β^{1} and the symmetry wall β^{2} − β^{1} (D = 6),

the magnetic wall β^{1} + β^{2} and the symmetry wall β^{3} − β^{2} (D = 7), and

the electric wall β^{1} + β^{2} + β^{3} and the symmetry wall β^{4} − β^{3} (D ≥ 8),
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 BKLlimit 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 KacMoody 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
5.2.1.2 Scale factor space and the Cartan subalgebra
For this reason, one can usefully identify the space of the scale factors with the Cartan subalgebra \(\mathfrak{h}\) of the KacMoody 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 \(\mathfrak{h} \subset \mathfrak{g}(A)\).
5.2.2 The fundamental Weyl chamber and the billiard table
5.2.3 Hyperbolicity implies chaos
If the billiard region of a gravitational system in the BKLlimit can be identified with the fundamental Weyl chamber of a hyperbolic KacMoody algebra, then the dynamics is chaotic.
5.2.3.1 Example: Pure gravity in D = 3 + 1 and \(A_1^{+ +}\)
Let us return once more to the example of pure fourdimensional 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 KacMoody 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.
5.3 Understanding the emerging KacMoody algebra
We shall now relate the KacMoody algebra whose fundamental Weyl chamber emerges in the BKLlimit to the Uduality 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 ChevalleySerre 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 ChevalleySerre 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 BKLlimit. By computing the billiard in three spacetime dimensions instead of in maximal dimension, the relation to Uduality 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 nongravitational 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 nonsplit, 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 elevendimensional supergravity on a circle.
For nontoroidal 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 lowerdimensional theory is modified. In many cases, the billiard of the effective compactified theory is described by a (nonhyperbolic) regular Lorentzian subalgebra of the original hyperbolic KacMoody algebra [98].
The walls are also invariant under dualization of a pform into a (D − p − 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 threedimensional reduced theory is described, after dualization of all vectors to scalars, by the sum of the EinsteinHilbert 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 Uduality group \({{\mathcal U}_3}\) and the KacMoody algebras appearing in the BKLlimit, we must first discuss some important features of the Lie algebra \(\mathfrak{u}_3\).
The Lagrangian (5.45) coupled to the pure threedimensional EinsteinHilbert term is the key to understanding the appearance of the Lorentzian Coxeter group \(\mathfrak{u}_3^{++}\) in the BKLlimit.
5.3.3 Starting at the bottom — Overextensions of finitedimensional Lie algebras
The structure of the corresponding Lorentzian KacMoody 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 E_{8}, the magnetic root of Equation (5.50) provides the affine extension, while the gravitational root of Equation (5.52) is the overextended root.

The dominant electric wall forms \({\hat e^{\hat \chi}}(\hat \varphi)\) for the supergravity theory in question are in onetoone correspondence with the simple roots of the associated Uduality 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 BKLanalysis of supergravity theories is a generic phenomenon closely linked to hidden symmetries.
5.4 Models associated with split real forms
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 KacMoody algebras that underlie the gravitational dynamics in the BKLlimit. These appear as overextensions of the finite Lie algebras found in three dimensions. Finally we indicate which of these theories are related to string/Mtheory.
\(\mathcal{U}_3/\mathcal{K}(\mathcal{U}_3)\)  Lagrangian in maximal dimension  KacMoody algebra  String/Mtheory 

\({{SL(n + 1,{\mathbb R})} \over {SO(n + 1)}}\)  \({\mathcal{L}}_{n+3} = R\star \bf{1}\)  \(AE_{n+2}\equiv A_n^{++}\)  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 
\({{{E_{8(8)}}} \over {{\rm{Spin(16)/}}{{\mathbb Z}_2}}}\)  \({{\mathcal L}_{11}} = R \star {\bf{1}}  {1 \over 2} \star d{C^{(3)}} \wedge d{C^{(3)}}  {1 \over 6}d{C^{(3)}} \wedge d{C^{(3)}} \wedge {C^{(3)}}\)  \({E_{10}} = E_8^{+ +}\)  Mtheory, 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 FiniteDimensional Real Lie Algebras
In this section we explain the basic theory of real forms of finitedimensional 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 KacMoody billiards for nonsplit 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 KacMoody algebras in Section 4, emphasizing here more involved features of the general structure theory of real Lie algebras rather than relying entirely on the ChevalleySerre 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 Uduality 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 finitedimensional 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 TitsSatake 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
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 semisimple 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 semisimple Lie algebras.
6.2.1 Real forms of \({\mathfrak {sl}}(2,\,{\mathbb C})\)
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 \(a({\mathfrak {h}_1}) = {\mathfrak {h}_2}\).
6.2.3 The Killing form
6.2.4 The compact real form \({\mathfrak {su}}(2)\)
The corresponding group obtained by exponentiation is SU(2), which is isomorphic to the 3sphere 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 A_{1} 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
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 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”.
6.2.6 Concluding remarks
 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.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.Any element of \({\mathfrak p}\) is conjugated via \({\mathcal K}\) to a multiple of h,so, denoting by \({\mathfrak a}={\mathbb R}h\) the (maximal) noncompact Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\), we obtain$$\rho (\cos \alpha h + \sin\alpha x) = \left({\begin{array}{*{20}c} {\cos {{\alpha} \over 2}} & {\sin{{\alpha} \over 2}}\\ { \sin{{\alpha} \over 2}} & {\cos {{\alpha} \over 2}} \end{array}} \right)\rho \,h\;\left({\begin{array}{*{20}c} {\cos {{\alpha} \over 2}} & { \sin{{\alpha} \over 2}}\\ {\sin{{\alpha} \over 2}} & {\cos {{\alpha} \over 2}} \end{array}} \right),$$(6.26)$${\mathfrak{p}} = {\rm{Ad}}({\mathcal K}){\mathfrak{a}}{.}$$(6.27)
 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.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 nonequivalent Cartan subalgebra by taking the combinationSimilarly, the normalized root vectors associated with t are (up to a complex phase) E_{±2i} = ½(h ∓ ix):$$t = e + \theta (e).$$(6.28)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$${[}t,\;{E_{2i}}] = 2i\,{E_{2i}},\qquad {[}t,\;{E_{ 2i}}] =  2i\;{E_{ 2i}},\qquad {[}{E_{2i}},\;{E_{ 2i}}] = i\;t.$$(6.29)$$\cos \alpha \; h + \sin\;\alpha \,x = {e^{i\alpha}}{E_{2i}} + {e^{ i\alpha}}{E_{ 2i}}.$$(6.30)
6.3 The compact and split real forms of a semisimple Lie algebra
We shall consider here only semisimple Lie algebras. Over the complex numbers, Cartan subalgebras 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 nondiagonalizable elements and would not be Cartan subalgebras^{23}.
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.
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 onetoone 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
Alignment is not automatic. For instance, one can always dealign 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
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 semisimple Lie algebra, one can show that they are conjugated by an internal automorphism [93, 129]. It follows that any real semisimple Lie algebra possesses a “unique” Cartan involution.
An important consequence of this [93, 129] is that any real semisimple 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 semisimple 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
One, obviously nonvanishing, subspace is g_{0}, 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 }] ⊂ g_{λ+μ}, while \({{\mathfrak a}_0} \subset {{\mathfrak p}_0}\) implies that θg_{ λ } = g_{−λ}. Thus if λ is a restricted root, so is −λ.
6.4.5 Iwasawa and \({\mathcal {KAK}}\) decompositions
The Iwasawa decomposition provides a global factorization of any semisimple Lie group in terms of closed subgroups. It can be viewed as the generalization of the GramSchmidt orthogonalization process.
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 socalled \({\mathcal K}{\mathcal A}{\mathcal K}\) 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 θstable Cartan subalgebras
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 θstable 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 θstable 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 noncompact imaginary roots
Consider a general θstable 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 nonzero eigenvalue of ad \({\mathfrak h}\) where \({\mathfrak h}\) is the complexification of the θstable 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 nonfinitedimensional KacMoody algebras.
6.4.8 Jumps between Cartan subalgebras — Cayley transformations
These two kinds of transformations — called Cayley transformations — allow, starting from a θstable 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
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 nonvanishing component along \(i{{\mathfrak t}_0}\), and the first nonzero 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 2tuples 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 2element orbits under θ are exhibited by joining the correponding simple roots by a double arrow and the 1element 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
6.5.1.1 \({\mathfrak {sl}}(5,\,{\mathbb R})\) and \({\mathfrak {su}}(5)\)
6.5.1.2 The other real forms
6.5.1.3 Vogan diagrams
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.
6.5.3 Cayley transformations in su(3, 2)
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 (γ∣α) > 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) = s_{θ(α)}(θ(β)) is also in Δ.
6.5.5 Illustrations: \({\mathfrak {sl}}(4,\,{\mathbb R})\) versus \({\mathfrak {sl}}(2,\,{\mathbb H})\)
6.5.6 A pictorial summary — All real simple Lie algebras (Vogan diagrams)
Vogan diagrams (A_{ n } series)
A_{ n } series, n ≥ 1  Vogan diagram  Maximal compact subalgebra 

\({\mathfrak s}{\mathfrak u}(n + 1)\)  \({\mathfrak s}{\mathfrak u}(n + 1)\)  
\({\mathfrak s}{\mathfrak u}(p,\,q)\)  \({\mathfrak s}{\mathfrak u}(p)\, \oplus \,{\mathfrak s}{\mathfrak u}(q)\, \oplus \,u(1)\)  
\({\mathfrak s}{\mathfrak l}(2n,\,{\mathbb R})\)  \({\mathfrak s}{\mathfrak o}(2n)\)  
\({\mathfrak s}{\mathfrak l}(2n + 1,\,{\mathbb R})\)  \({\mathfrak s}{\mathfrak o}(2n + 1)\)  
\({\mathfrak s}{\mathfrak l}(n + 1,\,{\mathbb H})\)  \({\mathfrak s}{\mathfrak p}(n + 1)\) 
Vogan diagrams (B_{ n } series)
B_{ n } series, n ≥ 2  Vogan diagram  Maximal compact subalgebra 

\({\mathfrak s}{\mathfrak o}(2n + 1)\)  \({\mathfrak s}{\mathfrak o}(2n + 1)\)  
\(\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak o}(p,\,q)} \\ {p \leq n  {1 \over 2},\,q = 2n + 1  p} \\ \end{array}\)  \({\mathfrak s}{\mathfrak o}(p)\, \oplus \,{\mathfrak s}{\mathfrak o}(q)\) 
Vogan diagrams (C_{ n } series)
C_{ n } series, n > 3  Vogan diagram  Maximal compact subalgebra 

\({\mathfrak s}{\mathfrak p}(n)\)  \({\mathfrak s}{\mathfrak p}(n)\)  
\(\begin{array}{*{20}c}{{\mathfrak s}{\mathfrak p}(p,\,q)} \\ {0 < p \leq {n \over 2},\,q = n  p} \\ \end{array}\)  \({{\mathfrak s}{\mathfrak p}(p)\, \oplus \,{\mathfrak s}{\mathfrak p}(q)}\)  
\({{\mathfrak s}{\mathfrak p}(n,\,{\mathbb R})}\)  \({{\mathfrak u}(n)}\) 
Vogan diagrams (D_{ n } series)
D_{ n } series, n ≥ 4  Vogan diagram  Maximal compact subalgebra 

\({\mathfrak s}{\mathfrak o}(2n)\)  \({\mathfrak s}{\mathfrak o}(2n)\)  
\(\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak o}(2p,\,2q)} \\ {0 < p \leq {n \over 2},\,q = n  p} \\ \end{array}\)  \({{\mathfrak s}{\mathfrak o}(2p)\, \oplus \,{\mathfrak s}{\mathfrak o}(2q)}\)  
\({{\mathfrak s}{\mathfrak o}\ast (2n)}\)  \({{\mathfrak u}(n)}\)  
\(\begin{array}{*{20}c} {{\mathfrak s}{\mathfrak o}(2p + 1,\,2q + 1)} \\ {0 < p \leq {{n  1} \over 2},} \\ {q = n  p  1} \\ \end{array}\)  \({\mathfrak s}{\mathfrak o}(2p + 1)\, \oplus \,{\mathfrak s}{\mathfrak o}(2q + 1)\) 
Vogan diagrams (G_{2} series)
G _{2}  Vogan diagram  Maximal compact subalgebra 

G _{2}  G _{2}  
G _{2(2)}  \({\mathfrak s}{\mathfrak u}(2)\, \oplus \,{\mathfrak s}{\mathfrak u}(2)\) 
Vogan diagrams (F_{4} series)
F_{4} series  Vogan diagram  Maximal compact subalgebra 

F _{4}  F _{4}  
F _{4(4)}  \({\mathfrak s}{\mathfrak p}(3)\, \oplus \,{\mathfrak s}{\mathfrak u}(2)\)  
F_{4(−20)}  \({\mathfrak s}{\mathfrak o}(9)\) 
Vogan diagrams (E_{6} series)
E _{6}  Vogan diagram  Maximal compact subalgebra 

E _{6}  E _{6}  
E _{6(6)}  \({\mathfrak s}{\mathfrak p}(4)\)  
E _{6(2)}  \({\mathfrak s}{\mathfrak u}(6)\, \oplus \,{\mathfrak s}{\mathfrak u}(2)\)  
E _{6(−14)}  \({\mathfrak s}{\mathfrak u}(10)\, \oplus \,{\mathfrak u}(1)\)  
E _{6(−26)}  F _{4} 
Vogan diagrams (E_{7} series)
E _{7}  Vogan diagram  Maximal compact subalgebra 

E _{7}  E _{7}  
E _{7(7)}  \({\mathfrak s}{\mathfrak u}(8)\)  
E _{7(43)}  \({\mathfrak s}{\mathfrak o}(12)\, \oplus \,{\mathfrak s}{\mathfrak u}(2)\)  
E _{7(} _{−} _{25)}  \({E_6}\, \oplus {\mathfrak u}(1)\) 
Vogan diagrams (E_{8} series)
E _{8}  Vogan diagram  Maximal compact subalgebra 

E _{8}  E _{8}  
E _{8(8)}  \({\mathfrak s}{\mathfrak o}(16)\)  
E _{8(} _{−} _{24)}  \({E_7}\, \oplus {\mathfrak s}{\mathfrak u}(2)\) 
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 noncompact 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 }  \({\mathfrak u}{\mathfrak s}{\mathfrak p}(2n)\) 
\({\mathfrak s}{\mathfrak l}(p + q,\,{\mathbb C})\)  −I_{p,q} X^{†} I_{p,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 }  \({\mathfrak s}{\mathfrak u}(n)\, \oplus \,{\mathfrak u}(1)\) 
\({\mathfrak s}{\mathfrak p}(p + q,\,{\mathbb C})\)  −K_{p,q} X^{†} K_{p,q}  L _{p,q} X ^{ t } L _{p,q}  \({\mathfrak s}{\mathfrak p}(p)\, \oplus \,{\mathfrak s}{\mathfrak p}(q)\) 
6.6 TitsSatake diagrams
The classification of real forms of a semisimple 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 semisimple Lie algebras, developed mainly by Araki [5], and based on socalled TitsSatake diagrams [161, 155].
6.6.1 Example 1: \({\mathfrak {su}}(3,\,2)\)
6.6.1.1 Diagonal description
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.
6.6.1.3 Restricted roots
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
6.6.2.2 Cartan involution and roots
6.6.2.3 Restricted roots
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 H_{4}) and contains more than just the three compact Cartan generators h_{2}, h_{3} and h_{4}. 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 α_{4}. The algebra m constitutes a rank 3, 9dimensional Lie algebra, which can be identified with \({\mathfrak {su}}(3) \oplus {\mathfrak u}(1)\).
6.6.3 TitsSatake diagrams: Definition
6.6.4 Formal considerations
TitsSatake diagrams provide a lot of information about real semisimple 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 black roots of a TitsSatake diagram represent B_{0} 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 semisimple compact Lie algebra and onedimensional, 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 semisimple part and of the number of \({{\mathfrak u}(1)}\) terms, while the dimension of \({\mathfrak m}\) is equal to the dimension of its semisimple part and of the number of \({{\mathfrak u}(1)}\) terms. The Dynkin diagram of \({\mathfrak m}\) reduces to the Dynkin diagram of its semisimple part.
6.6.5 Illustration: F_{4}
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 semisimple 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.
6.7 The real semisimple 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 TitsSatake diagrams.
6.7.1 Dimensions l = 2q +1 < k = 2p
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(p − q) − 1. These constitute a B_{2q+1} root system.
6.7.2 Dimensions l = 2q + 1 > k = 2p
The restricted root system is now of type B_{2p}, with 4p(2p − 1) long roots of multiplicity one and 4p short roots of multiplicity 2(q − p) + 1.
6.7.3 Dimensions l = 2q, k = 2p
 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)
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(p − q). 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 — TitsSatake diagrams for noncompact real forms
To summarize the analysis, we provide the TitsSatake diagrams for all noncompact real forms of all simple Lie algebras [5, 93]. We do not give explicitly the TitsSatake diagrams of the compact real forms as these are simply obtained by painting in black all the roots of the standard Dynkin diagrams.

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), G_{2}, F_{4}, E_{6}, E_{7}, or E_{8}, 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).
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  

\({\mathfrak s}{\mathfrak u}(p,\,q)\quad p \geq q > 0p + q \geq 2\)  p ≥ q > 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 } 
p ≥ q > 0 p + q = 2n ≥ 8  q  B_{ q } if p > q, D_{ q } if p = q  
\({\mathfrak s}{\mathfrak p}(p,\,q)\quad p \geq q > 0p + q \geq 3\)  p ≥ q > 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} 
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 KacMoody Billiards II — The Case of NonSplit Real Forms
We will now make use of the results from the previous section to extend the analysis of KacMoody billiards to include also theories whose Uduality symmetries are described by algebras \(\mathfrak{u}_3\) that are nonsplit. 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”
As explained in Section 4.9.2, the restricted root system of the real form \(\mathfrak{u}_{3}\) can be either reduced or nonreduced. If it is reduced, it corresponds to one of the root systems of the finitedimensional simple Lie algebras. On the other hand, if the restricted root system is nonreduced, it is necessarily of (BC)_{ n }type [93] (see Figure 19 for a graphical presentation of the BC_{3} 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, semisimple, 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.
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 Uduality 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 restricted root system Σ of u_{3} 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 }, G_{2}, F_{4}, E_{6}, E_{7} or E_{8}.

The restricted root system, Σ, of \(\mathfrak{u}_3\) is of nonreduced 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 fourdimensional gravity coupled to a Maxwell field. This is the bosonic sector of \({\mathcal N} = 2\) supergravity in four dimensions, which has the nonsplit real form \(\mathfrak{su}(2,\,1)\) as its Uduality symmetry. The restricted root system of \(\mathfrak{su}(2,\,1)\) is the nonreduced (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 Uduality algebra is thus \(\mathfrak{so}(8,\,24)\), which is nonsplit. In this example, however, the restricted root system is B_{8}, 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.
TitsSatake diagrams (A_{ n } series)
A_{ n } series n ≥ 1  TitsSatake diagram  Restricted root system 

\({{\mathfrak s}{\mathfrak l}(n,\,{\mathbb R}),\,n \geq 3}\)  
A I  A _{ n }  
\({{\mathfrak s}{\mathfrak u}\ast(n + 1),\,n = 2k + 1}\)  
A II  A _{2k}  
\({{\mathfrak s}{\mathfrak u}(p,\,n + 1  p)}\)  
A III  BC _{ p }  
\({{\mathfrak s}{\mathfrak u}\left({{{n + 1} \over 2},\,{{n + 1} \over 2}} \right),\,n = 2k + 1}\)  
A III  C _{(k+1)}  
\({\mathfrak s}{\mathfrak u}(1,\,n  1),\,n \geq 3\)  
A IV  A _{1} 
TitsSatake diagrams (B_{ n } series)
B_{ n } series n ≥ 4  TitsSatake diagram  Restricted root system 

\({{\mathfrak s}{\mathfrak o}(p,\,2n  p + 1),\,p \geq 1}\)  
B I  B _{ p }  
\({{\mathfrak s}{\mathfrak o}(1,\,2n)}\)  
B II  A _{1} 
TitsSatake diagrams (C_{ n } series)
C_{ n } series n ≥ 3  TitsSatake diagram  Restricted root system 

\({{\mathfrak s}{\mathfrak p}(n,\,{\mathbb R})}\)  
C I  C _{ n }  
\({{\mathfrak s}{\mathfrak p}(p,\,n  p)}\)  
C II  B _{ p }  
\({{\mathfrak s}{\mathfrak p}\left({{n \over 2},\,{n \over 2}} \right),\,n = 2k}\)  
C II  \({C_{{n \over 2}}}\) 
TitsSatake diagrams (D_{ n } series)
D_{ n } series n ≥ 4  TitsSatake diagram  Restricted root system 

\({{\mathfrak s}{\mathfrak o}(p,\,2n  p),\,p \leq n  2}\)  
D I  B _{ p }  
\({{\mathfrak s}{\mathfrak o}(n  1,\,n + 1)}\)  
D I  B _{(n−1)}  
\({{\mathfrak s}{\mathfrak o}(n,\,n)}\)  
D I  D _{ n }  
\({{\mathfrak s}{\mathfrak o}(1,\,2n  1)}\)  
D II  A _{1}  
\({{\mathfrak s}{\mathfrak o}(\ast(2n)),\,n = 2k}\)  
D III  C _{2k−1}  
\({{\mathfrak s}{\mathfrak o}(\ast(2n)),\,n = 2k + 1}\)  
D III  BC _{2k} 
TitsSatake diagrams (G_{2} series)
G_{2} series  TitsSatake diagram  Restricted root system 

G _{2(2)}  
G 
TitsSatake diagrams (F_{4} series)
F_{4} series  TitsSatake diagram  Restricted root system 

F _{4(4)}  
F I  
F _{4(− 20)}  
F II 
TitsSatake diagrams (E_{6} series)
E_{6} series  TitsSatake diagram  Restricted root system 

E _{6(6)}  
E I  
E _{6(2)}  
E II  
E _{6(−14)}  
E III  
E _{6(− 26)}  
E IV 
TitsSatake diagrams (E_{7} series)
E_{7} series  TitsSatake diagram  Restricted root system 

E _{7(7)}  
E V  
E _{7(− 5)}  
E VI  
E _{7(− 25)}  
E VII 
TitsSatake diagrams (E_{8} series)
E_{8} series  TitsSatake diagram  Restricted root system 

E _{8(8)}  
E VIII  
E _{8(− 24)}  
E IX 
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 BKLlimit the dynamics of the coupled EinsteinMaxwell system in fourdimensions is equivalent to that of pure fourdimensional 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 KacMoody algebra \(D{E_{10}} = D_8^{+ +}\) [45]. This algebra is the overextension of the Uduality 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 D_{8}, so we have \({\mathfrak f} = {{\mathfrak u}_3}\).
By adding one Maxwell field to the theory we modify the billiard to the hyperbolic KacMoody algebra \(B{E_{10}} = B_8^{+ +}\), which is the overextension of the split form \({\mathfrak {so}}(8,\,9)\) of B_{8} [45]. This is the case relevant for (the bosonic sector of) Type I supergravity in ten dimensions. In both these cases the relevant KacMoody 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 Uduality algebra is nonsplit. For the heterotic string, the bosonic field content of the corresponding supergravity is given by pure gravity coupled to a dilaton, a 2form and an E_{8} × E_{8} YangMills 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 tendimensional theory discussed above. Geometrically, these 16 Maxwell fields correspond to the KaluzaKlein vectors arising from the compactification on T^{16} of the 26dimensional bosonic leftmoving sector of the heterotic string [89].
Consequently, the relevant Uduality algebra is \({\mathfrak {so}}(8,\,8+16)={\mathfrak {so}}(8,\,24)\) which is a nonsplit real form. But we know that the billiard for the heterotic string is governed by the same KacMoody 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 oneforms are degenerate 16 times. We now want to understand this apparent discrepancy using the machinery of nonsplit real forms exhibited in previous sections. The same discussion applies to the SO(32)superstring.
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 B_{2q}. 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) \({\Sigma _{{\mathfrak {so}}(8,\,24)}} = {B_8}\).
This analysis again showed explicitly how it is always the split symmetry that controls the chaotic behavior in the BKLlimit. It is important to point out that when going beyond the strict BKLlimit, 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 nonsplit real forms
In this section we provide a list of all theories coupled to gravity which, upon compactification to three dimensions, display Uduality 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 pforms can be dualized to scalars or to 1forms 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 fourdimensional 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 nonsplit cases is the possible appearance of socalled twisted overextensions. These arise when the restricted root system of \({\mathcal U}_3\) is of nonreduced type hence yielding a twisted affine KacMoody 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 nonreduced ((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 Uduality 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 Uduality 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 BKLlimit also for this theory.
The case of \({\mathcal N} = 5\) follows along similar lines. In D = 3 the nonsplit form E_{6(−14)} of E_{6} appears, whose maximal split subalgebra is \({\mathfrak f} = {C_2}\). However, the relevant KacMoody algebra is not \(C_2^{+ +}\) but rather \(A_4^{(2) +}\) because the restricted root system of E_{6(−14)} is (BC)_{2}.
Classification of theories whose Uduality symmetry in three dimensions is described by a nonsplit 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 KacMoody 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 KacMoody 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 KacMoody algebras naturally emerge when analysing gravity in the extreme BKL regime. This has led to the conjecture that the corresponding KacMoody 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 KacMoody 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 infinitedimensional structure of a Lorentzian KacMoody algebra \({\mathfrak g}\). In this section we present such a method.
The method by which we shall deal with the infinitedimensional structure of a Lorentzian KacMoody algebra \({\mathfrak g}\) is based on a certain gradation of \({\mathfrak g}\) into finitedimensional subspaces \({{\mathfrak g}_\ell}\). More precisely, we will define a socalled level decomposition of the adjoint representation of \({\mathfrak g}\) such that each level ℓ 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 ℓ (see Section 9).
We begin by illustrating these ideas for the finitedimensional 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 finitedimensional example: \({\mathfrak {sl}(3,\,\mathbb R)}\)
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 ChevalleySerre presentation as for \({\mathfrak {sl}(3,\,\mathbb C)}\), but with all coefficients restricted to the real numbers.
We can then conclude that \(\Lambda _{{\rm{ad}}}^{(0)} =  {\alpha _1}\) is the lowest weight of the threedimensional adjoint representation 3_{0} of \({\mathfrak {sl}(2,\,\mathbb R)}\) with weights \(\{\Lambda _{{\rm{ad}}}^{(0)},\,0,\,  \Lambda _{{\rm{ad}}}^{(0)}\}\), where the subscript on 3_{0} again indicates that this representation is located at level ℓ = 0 in the decomposition. The module for this representation is \(\mathcal L(\Lambda _{{\rm{ad}}}^{(0)}) = span\{{f_1},\,\alpha _1^ \vee, \,{e_1}\}\).
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 ℓ = 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.
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 KacMoody algebra with respect to an arbitrary number s < r of the simple roots and then the level ℓ is generalized to the “multilevel” ℓ = (ℓ_{1}, ⋯, ℓ_{ s }).
8.2.1 Gradation
We consider a KacMoody 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 \({\mathfrak g}\).
At level zero, ℓ = (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 ℓ_{ a }’s is positive, all the other ones must be nonnegative 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 \({\mathfrak r}\)
8.2.3 Outer multiplicity
There is an interesting relationship between root multiplicities in the KacMoody 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 KacMoody 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).

The multiplicity mult(γ) of each \(\gamma \in {\mathfrak h}_{\mathfrak g}^{\star}\) at level ℓ 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 ℓ 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.)
Multiplicities m_{ α } = mult(α) and comultiplicities c_{ α } of all roots α of AE_{3} up to height 10.
ℓ  m _{1}  m _{2}  c _{ α }  m _{ α }  α ^{2} 

0  0  1  1  1  2 
0  0  k > 1  1/k  0  2k^{2} 
0  1  0  1  1  2 
0  k > 1  1  1/k  0  2k^{2} 
1  0  0  1  1  2 
k > 0  0  0  1/k  0  2 k^{2} 
0  1  1  1  1  2 
0  k > 1  k > 1  1/k  0  2 k^{2} 
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  1  2 
4  2  2  1/2  0  8 
1  2  2  1  1  2 
2  4  4  1/2  0  8 
2  2  1  2  2  −2 
4  4  2  8  7  −8 
2  3  0  1  1  2 
4  6  0  1/2  0  8 
3  2  0  1  1  2 
6  4  0  1/2  0  8 
2  3  1  2  2  −2 
3  2  1  1  1  0 
2  4  1  1  1  2 
2  3  2  2  2  −2 
3  2  2  1  1  2 
3  3  1  3  3  −4 
3  4  0  1  1  2 
4  3  0  1  1  2 
2  3  3  1  1  2 
3  4  1  3  3  −4 
2  4  3  1  1  2 
3  3  2  3  3  −4 
4  3  1  2  2  −2 
3  4  2  5  5  −6 
3  5  1  1  1  0 
4  3  2  2  2  −2 
4  4  1  5  5  −6 
4  5  0  1  1  2 
5  4  0  1  1  2 
3  4  3  3  3  −4 
3  5  2  3  3  −4 
4  3  3  1  1  2 
4  5  1  5  5  −6 
5  4  1  3  3  −4 
8.3 Level decomposition of AE_{3}
The KacMoody 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 KacMoody algebras. From a physical point of view, it is the Weyl group of AE_{3} which governs the chaotic behavior of pure fourdimensional gravity close to a spacelike singularity [46], as we have explained. Moreover, as we saw in Section 3, the Weyl group of AE_{3} is isomorphic with the wellknown arithmetic group \(PGL(2,\,{\mathbb Z})\) which has interesting properties [75].
From previous sections we recall that AE_{3} 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 ℓ = 0. The lightcone separates real roots from imaginary roots and so it is clear that if a representation at some level ℓ 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 ℓ = 0
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
The module for the representation 8_{0} is realized by the eight traceless generators \({K^i}_j\) of \(\mathfrak{sl}(3,\, \mathbb{R})\) and the module for the representation 1_{0} corresponds to the “trace” \(\alpha _1^ \vee\).
Note that the highest weight Λ_{hw} of a given representation of \(\mathfrak{r}\) is not in general equal to minus the lowest weight Λ of the same representation. In fact, −Λ_{hw} 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 selfconjugate that we have Λ_{hw} = −Λ. This is the case for example in the adjoint representation 8_{0}.
It is interesting to note that since the weights of a representation at level ℓ are related by Weyl reflections to weights of a representation at level −ℓ, it follows that the negative of a lowest weight Λ^{(ℓ)} at level ℓ is actually equal to the highest weight \(\Lambda _{{\mathrm{hw}}}^{( \ell)}\) of the conjugate representation at level −ℓ. Therefore, the Dynkin labels at level ℓ as defined here are the standard Dynkin labels of the representations at level −ℓ.
8.3.3 Level ℓ = 1
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].
8.3.5 Level ℓ = 2
Moreover, the representation [p_{2},p_{3}] = [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 sixdimensional and has highest weight 2λ_{3}, 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 AE_{3}. Therefore we know that it has root multiplicity one and may therefore only occur once in the level decomposition. Since the weight 2λ_{3} already appears in the larger representation 15_{2} it cannot be a highest weight in another representation at this level. Hence, the representation [0, 2] is not allowed within AE_{3}. A similar analysis reveals that also the representation [p_{2},p_{3}] = [1, 0], although allowed by Equation (8.54), has vanishing outer multiplicity.
8.3.6 Level ℓ = 3
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 ℓ = 4
8.4 Level decomposition of E_{10}
As we have seen, the KacMoody algebra E_{10} is one of the four hyperbolic algebras of maximal rank, the others being BE_{10}, DE_{10} and CE_{10}. It can be constructed as an overextension of E_{8} and is therefore often denoted by \(E_8^{+ +}\). Similarly to E_{8} in the rank 8 case, E_{10} is the unique indefinite rank 10 algebra with an even selfdual root lattice, namely the Lorentzian lattice Π_{1,9}.
Our first encounter with E_{10} in a physical application was in Section 5 where we have showed that the Weyl group of E_{10} describes the chaos that emerges when studying elevendimensional supergravity close to a spacelike singularity [45].
Moreover, decomposing E_{10} 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 α_{1} [125]. Similarly the IIBfields 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 α_{2} [126]. The extra A_{1}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 \(\mathfrak{sl}(10,\, \mathbb{R})\)
The representation content at each level is represented by \(\mathfrak{sl}(10,\, \mathbb{R})\)tensors whose index structure are encoded in the Dynkin labels [p_{1}, ⋯,p_{9}]. At level ℓ = 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.
The lowlevel representations in a decomposition of the adjoint representation of E_{10} into representations of its A_{9} subalgebra obtained by removing the exceptional node in the Dynkin diagram in Figure 49.
ℓ  Λ^{(ℓ)} = [p_{1}, ⋯, p_{9}]  Λ^{(ℓ)} = (m_{1}, ⋯, m_{10})  A_{9}representation  E_{10}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
8.4.2 “Gradient representations”
So far, we have only discussed the representations occurring at the first four levels in the E_{10} 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 Taylorlike 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.