An abstract setting for hamiltonian actions

Abstract

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