Which Lie algebra admit symplectic forms

Question

Hello, I am interested in symplectic Lie groups, I will sketch out their definition. A symplectic Lie group is a given pair $(G,\omega)$, where $G$ is a Lie group and $\omega$ is a left invariant symplectic form on $G$.

The algebraic situation is as follows. If $\mathcal{G}$ is the Lie algebra of $G$ and $\omega$ is a non-degenerate $2$-cocycle on $\mathcal{G}$, that is $\omega\in\wedge^2\mathcal{G}^*$ such that for any $x,y,z\in\mathcal{G}$ $$\omega([x,y],z)+\omega([y,z],x)+\omega([z,x],y)=0$$ The left invariant $2$-form associated to $\omega$ is a symplectic form on $G$.

My questions are:

1) What are the Lie algebras that admit such $omega$? The algebra can not be semi-simple, according to an article by Chu, Symplectic homogeneous spaces.

2) What is the link with the existence of an invertible solution of the classical Yang-Baxter equation $[r,r]=0$ for $r\in\wedge^2\mathcal{G}$ where $[\,,\,]$ is the Schouten bracket?

Thanks for any help.

Answer 1

Hello Amine,

One of the most general facts is that, if a Lie group admits a torsion free and (locally) flat connection (some authors call it an affine structure) which is invariant under left translations, then its Lie algebra properly contains (is not equal to) its derived ideal (Lie ideal generated by the Lie bracket of all vectors.) Equivalently, the dimension of such a Lie group is strictly greater than that of its commutator subgroup. As a particular case, this obstruction bans semisimple Lie groups from being endowed with such connections.

That is a result of J. Helmstetter, in "Radical d’une algebre symetrique a gauche. Ann. Inst. Fourier 29, no 4, (1979), 17-35.

Now, it is well known since the end of the 60s, that a left invariant symplectic structure in a Lie group provides the latter with a left invariant flat and torsion free connection, just as above (Koszul, Vinberg, Matsushima,...). If $\omega$ is the corresponding left invariant symplectic form, then $\nabla$ is provided by the formula $\omega (\nabla_xy,z):= - \omega (y, [x,y])$, where, of course, $\nabla$ stands for the needed connection and $x,y,z$ are any left invariant vector fields on the Lie group. Hence the same obtruction obviously applies, as well.

However, there are many nonsolvable symplectic Lie groups (Lie groups with a left invariant symplectic structure) and the most discussed amongst them, is certainly the ordinary group of affine transformations of the Euclidean space of dimension strictly greater than 1. Here the expression "affine transformations" relates to the ordinary ones, i.e. to transformations of the form F(x) = L(x) + v. Where L is a linear nonsingular map and v is a constant vector.

As for the relationship with the Classical Yang-Baxter Equation (CYBE), to the more general answer provided above by Nicola Ciccoli (D.V Alekseevsky and A.M. Perelomov, Journal of Geometry and Physics 22 (1997) 191-211, or, A. Lichnerowicz and A. Medina, Letters Math. Phy. 16, 225-335, (1988), ... ), I must add that invariant symplectic structures (1-1) correspond to invertible (skew-symmetric) solutions of the CYBE on Lie groups (same construction as explained by Nicola Ciccoli). For references on this part, you can have a look at, for example, A. Diatta and A. Medina, Manuscripta Math. 114 (2004), no. 4, 477–486.

Now, as for the question as to which Lie groups can bear left invariant symplectic structures and for more general results (including general properties, relationship with homogeneous bounded domains or Kahler Geometry, existence questions, construction procedures such as the so-called double extention and T*-extension,...) I can refer you to various papers by A. Medina, A. Lichnerowicz and A. Medina, A. Medina and Ph. Revoy, J.M. Dardie and A. Medina, M. Nguiffo Boyom, M. Bordemann, I. Bajo, S. Benayadi, D.V. Alekseevsky, E.B. Vinberg, S.G. Gindikin, I.I. Pjateckii-Sapiro, J.Dorfmeister, K.Nakajima, ... amongst the loads of authors who have worked on various aspects of the subject. The litterature is quite rich and yet, several interesting deep questions are still waiting for answers. However, there is a complete classification in low dimension up to dimension 4 and, at least partly, in dimension 6.

Answer 2

Given a solution $r$ of the classical Yang-baxter, identified with a map $\mathfrak g^*\to \mathfrak g$ one has that $ im r$ is a Lie subalgebra on which $r^{-1}$ gives a symplectic Lie algebra strutture and viceversa. This is simply explained in Alekseevsky Peremolov Poisson and symplectic structures on Lie algebras.

Answer 3

It's a bit late now, but there is also a more recent and very exhaustive paper by Cortés and Baues which adresses and summarizes many related questions: http://arxiv.org/abs/1307.1629

Answer 4

Semi-simple Lie groups do not admit left invariant torsion free flat connections, the proof is 'easy', see:

H. Matsushima & K. Okamoto, Non-existence of torsion free flat connections on a real semisimple Lie group, Hiroshima Math. J. 9, 59-60 (1979). Zbl 0411.53006.

A Lie group endowed with a complete left invariant torsion free flat connection is necessary solvable. See:

D. Segal, The structure of complete left-symmetric algebras, Math. Ann. 293, No. 3, 569-578 (1992). Zbl 0766.17005.