TITLE: Which Lie algebra admit symplectic forms
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: 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