An abstract setting for hamiltonian actions

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 ${\mathfrak h}$ with values in an ${\mathfrak h}$ -module V, we associate subalgebras ${\mathfrak {sp}(\mathfrak h,\omega) \supseteq \mathfrak {ham}(\mathfrak h,\omega)}$ of symplectic, resp., hamiltonian elements. Then ${\mathfrak {ham}(\mathfrak h,\omega)}$ has a natural central extension which in turn is contained in a larger abelian extension of ${\mathfrak {sp}(\mathfrak h,\omega)}$ . In this setting, we study linear actions of a Lie group G on V which are compatible with a homomorphism ${\mathfrak g \to \mathfrak {ham}(\mathfrak h,\omega)}$ , i.e., abstract hamiltonian actions, corresponding central and abelian extensions of G and momentum maps ${J : \mathfrak g \to V}$ .