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{\mathfrak h} with values in an \mathfrak h{\mathfrak h}-module V, we associate subalgebras \mathfrak sp(\mathfrak h,w) ê \mathfrak ham(\mathfrak h,w){\mathfrak {sp}(\mathfrak h,\omega) \supseteq \mathfrak {ham}(\mathfrak h,\omega)} of symplectic, resp., hamiltonian elements. Then \mathfrak ham(\mathfrak h,w){\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,w){\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 ? \mathfrak ham(\mathfrak h,w){\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 ? V{J : \mathfrak g \to V}.