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}.
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