Homogeneous Symplectic Spaces and Central Extensions

TL;DR

This paper revisits Kirillov's classical result on homogeneous symplectic spaces, extending it to non-simply-connected groups using Neeb's theorem, and explores the relationship between central extensions and Lie algebra cocycles.

Contribution

It generalizes Kirillov's theorem to non-simply-connected groups by applying Neeb's theorem, clarifying the connection between central extensions and Lie algebra cohomology.

Findings

Kirillov's theorem extended to non-simply-connected groups

Neeb's theorem relates integrability of central extensions to Lie algebra cocycles

Homogeneous symplectic spaces are covers of coadjoint orbits of extended groups

Abstract

We summarise recent work (arXiv:2203.07405 [math.SG]) on the classical result of Kirillov that any simply-connected homogeneous symplectic space of a connected group $G$ is a hamiltonian $\widehat{G}$-space for a one-dimensional central extension $\widehat{G}$ of $G$, and is thus (by a result of Kostant) a cover of a coadjoint orbit of $\widehat{G}$. We emphasise that existing proofs in the literature assume that $G$ is simply-connected and that this assumption can be removed by application of a theorem of Neeb. We also interpret Neeb's theorem as relating the integrability of one-dimensional central extensions of Lie algebras to the integrability of an associated Chevalley--Eilenberg 2-cocycle.