Symplectic Actions and Central Extensions

Abstract

We give a proof of the fact that a simply-connected symplectic homogeneous space $(M,\omega )$ of a connected Lie group $G$ is the universal cover of a coadjoint orbit of a one-dimensional central extension of $G$. We emphasise the rôle of symplectic group cocycles and the relationship between such cocycles, left-invariant presymplectic structures on $G$ and central extensions of $G$; in particular, we show that integrability of a central extension of $\mathfrak{g}$ to a central extension of $G$ is equivalent to integrability of a representative Chevalley-Eilenberg 2-cocycle of $\mathfrak{g}$ to a symplectic cocycle of $G$.

23 pages, 2 appendices; Section 5.5 added and Appendix A expanded in v2

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