Abstract
In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky’s double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang–Baxter equation. In this context, we prove a factorization theorem for group-like elements. An explicit exponential solution of the corresponding Lie bracket flow is presented, which is based on the aforementioned post-Lie Magnus expansion.
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Notes
- 1.
Brainstorming Workshop on “New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series”, May 25-28, 2015, ICMAT, Madrid, Spain. Supported by a grant from Iceland, Liechtenstein and Norway through the EEA Financial Mechanism as well as the project “Mathematical Methods for Ecology and Industrial Management” funded by Ayudas Fundación BBVA a Investigadores, Innovadores y Creadores Culturales.
- 2.
In the following we will use \(*\)-notation to denote the differential of a smooth application \(\phi :M_1\rightarrow M_2\), if \(M_2\ne \mathbb F\). More precisely the differential of \(\phi \) at \(m\in M_1\) will be denoted as \(\phi _{*,m}\). Recall that this is a linear map between \(T_mM_1\) and \(T_{\phi (m)}M_2\) such that \((\phi _{*,m}v)f=v(f\circ \phi )\), for all \(v\in T_mM_1\) and all \(f\in C^\infty (M_2)\). On the other hand, if \(M_2=\mathbb F\), i.e., if \(\phi =H:M\rightarrow \mathbb F\) is a smooth function, we will write its differential at the point \(m\in M\) as \(dH_m\). Note that \(dH_m\in T^*_mM\).
- 3.
Let \((\mathscr {I},\le )\) be a directed poset. Recall that a pair \((\{A_i\}_{i\in \mathscr {I}},\{f_{ij}\}_{i,j\in \mathscr {I}})\) is called an inverse or projective system of sets over \(\mathscr {I}\), if \(A_i\) is a set for each \(i\in \mathscr {I}\), \(f_{ij}:A_i\rightarrow A_j\) is a map defined for all \(j\le i\) such that \(f_{ij}\circ f_{jk}=f_{ik}:A_i\rightarrow A_k\), every time the corresponding maps are defined and \(f_{ii}={\text {id}}_{A_i}\). Then the inverse or the projective limit of the inverse system \((\{A_i\}_{i\in \mathscr {I}},\{f_{ij}\}_{i,j\in \mathscr {I}})\) is
$$\begin{aligned} \varprojlim A_i=\{\xi \in \prod _{i\in \mathscr {I}}A_i\,\vert \, f_{ij}(p_i(\xi ))=p_j(\xi ),\,\forall j\le i\}, \end{aligned}$$where, for each \(i\in \mathscr {I}\), \(p_i:\prod _{i\in \mathscr {I}} A_i\rightarrow A_i\) is the canonical projection. This definition is easily specialized to define the inverse limit in the category of algebras, co-algebras and Hopf algebras.
- 4.
Recall that if M is a \(\mathbb Z\)-module endowed with a decreasing filtration, \(M=M_0\supset M_1\supset M_2\supset \cdots \), then a sequence \((x_k)_{k\in \mathbb N}\) is called a Cauchy sequence if for each r there exists \(N_r\), such that, if \(n,m>N_r\), then \(x_n-x_m \in M_r\). This amounts to saying, that if n, m are sufficiently large, then \(x_n+M_r=x_m+M_r\). This implies that \((x_k)_{k\in \mathbb N}\) is a coherent sequence, i.e., it belongs to \(\hat{M}=\varprojlim M/M_k\). In other words, every Cauchy sequence is convergent in \(\hat{M}\). These considerations can be extended verbatim to the case of complete augmented algebras.
- 5.
Formula (36) is known as the Bianchi’s 1st identity. Among many other identities fulfilled by the covariant derivatives of the torsion and curvature of a linear connection, the so called Bianchi’s 2nd identity is worth to recall:
$$ \sum _{\circlearrowleft }\big ((\nabla _X \mathrm {R})(Y,Z)+\mathrm {R}(\mathrm {T}(X,Y),Z)\big ) =0,\quad \forall X,Y,Z\in \mathfrak X_M. $$.
- 6.
One can define the notion of a Poisson algebra in a more algebraic setting, without any reference to some underlying manifold. More precisely one say that a associative and commutative algebra \((A,\cdot )\) is a Poisson algebra if there exists a \(\mathbb F\)-bilinear, skew-symmetric map \(\{\cdot ,\cdot \}:A\otimes _\mathbb F A\rightarrow A\), which is a bi-derivation of \((A,\cdot )\) and which fulfills the Jacobi identity. Such a bilinear map is called a Poisson bracket.
- 7.
The bracket \([\cdot ,\cdot ]_{SN}\) just introduced extends to \(\varLambda ^\bullet TP\), the full exterior algebra of TP, and is called the Schouten–Nijenhuis bracket.
- 8.
The symplectic foliation of a Poisson manifold \((P,\{\cdot ,\cdot \})\) is the generalized distribution in the sense of Sussmann defined on P by the Hamiltonian vector fields. Each leaf of this distribution is an immersed symplectic manifold, i.e., it is an immersed submanifold of P which carries a symplectic structure. See Example 7, which is defined by the restriction to the leaf of the Poisson structure.
- 9.
Note that this is not a restriction, since, by the Ado’s theorem, every finite dimensional Lie algebra admits a faithful finite dimensional representation.
- 10.
Recall that a quadratic Lie algebra \((\mathfrak g,B)\) is a Lie algebra endowed with a non-degenerate, \(\mathfrak g\)-invariant bilinear form \(B: \mathfrak g\otimes \mathfrak g \rightarrow \mathfrak g\), i.e., B is a bilinear form such that (1) if \(x \in \mathfrak g\) is such that \(B(x,y)=0\) for all \(y \in \mathfrak g\), then \(x=0\) and 2) \(B([x,y],z)+B(y,[x,z])=0\) for all \(x,y,z\in \mathfrak g\).
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Acknowledgements
The first author acknowledges support from the Spanish government under the project MTM2013-46553-C3-2-P.
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Ebrahimi-Fard, K., Mencattini, I. (2018). Post-Lie Algebras, Factorization Theorems and Isospectral Flows. In: Ebrahimi-Fard, K., Barbero Liñán, M. (eds) Discrete Mechanics, Geometric Integration and Lie–Butcher Series. Springer Proceedings in Mathematics & Statistics, vol 267. Springer, Cham. https://doi.org/10.1007/978-3-030-01397-4_7
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