Homogeneous Symplectic Spaces and Central Extensions

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 $\hat{G}$-space for a one-dimensional central extension $\hat{G}$ of $G$, and is thus (by a result of Kostant) a cover of a coadjoint orbit of $\hat{G}$. 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.

7 pages, submitted to International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, IHP, Paris, July 18-22, 2022

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• Homogeneous Symplectic Spaces and Central Extensions

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• Homogeneous Symplectic Spaces and Central Extensions