Assume that $G$ is a Lie group and at the same time it admits a symplectic structure.



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> Does $G$ necessarily admit a symplectic structure such that the right multiplication preserves the symplectic structure?
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Symplectic Lie groups
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](http://arxiv.org/abs/1307.1629)) for references), so $H$ has no such structure.


