An abstract setting for hamiltonian actions

Details An abstract setting for hamiltonian actions An abstract setting for hamiltonian actions

2008-02-22

arxiv.org/abs/0802.3360v1

Monatshefte fuer Mathematik, 159 (2010) 261-288

Mathematics

Symplectic Geometry

35 pages

Scientific paper

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

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