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 ${J : mathfrak g to V}$ .