An abstract setting for hamiltonian actions

TL;DR

This paper develops an abstract framework for Hamiltonian group actions using Lie algebra cohomology, central extensions, and momentum maps, generalizing classical symplectic geometry concepts.

Contribution

It introduces a novel abstract setup for Hamiltonian actions based on Lie algebra 2-cochains and explores associated extensions and momentum maps.

Findings

Defined subalgebras of symplectic and Hamiltonian elements

Constructed natural central and abelian extensions of Lie groups

Analyzed compatible linear actions and momentum maps

Abstract

In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain $\omega$ on a Lie algebra $h$ with values in an $h$-module $V$, we associate subalgebras $sp(h,\omega) \supeq ham(h,\omega)$ of symplectic, resp., hamiltonian elements. Then $ham(h,\omega)$ has a natural central extension which in turn is contained in a larger abelian extension of $sp(h,\omega)$. In this setting, we study linear actions of a Lie group $G$ on $V$ which are compatible with a homomorphism $g \to ham(h,\omega)$, i.e. abstract hamiltonian actions, corresponding central and abelian extensions of $G$ and momentum maps $J : g \to V$.