TITLE: Symplectic Lie groups
QUESTION [10 upvotes]: Assume that $G$ is a Lie group and at the same time it admits a symplectic structure.

Does $G$ necessarily admit a symplectic structure such that the right multiplication preserves the symplectic structure?

REPLY [12 votes]: Let $H$ be the universal covering of $\mathrm{SL}_2(\mathbf{R})\times\mathbf{R}$. Then $H$ is diffeomorphic to $\mathbf{R}^4$ and hence has a symplectic structure (as a manifold). However, every Lie group with a right-invariant symplectic structure is solvable (see Baues-Cortès (arXiv link) for references), so $H$ has no such structure.