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Moments and reduction for symplectic groupoids. (English) Zbl 0659.58016
A Hamiltonian action of a Lie group G on a symplectic manifold M is generated by a momentum mapping J: \(M\to {\mathfrak g}^*\) which is equivariant with respect to the coadjoint representation. (\({\mathfrak g}^*\) is the space dual to the Lie algebra \({\mathfrak g}\) of G.) A reduction procedure of interest in Hamiltonian mechanics consists of forming the quotient \(M_{\mu}=J^{-1}(\mu)/G_{\mu}\) where \(\mu\) is an element of \({\mathfrak g}^*\) and \(G_{\mu}\) is its coadjoint isotropy group. The purpose of this paper is to extend this reduction procedure to actions by a symplectic groupoid. The reduction procedure, commonly thought of as inherently “symplectic”, is seen here to have a purely algebraic expression in the context of groupoid actions.
Reviewer: W.Satzer

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H30 Other variational principles in mechanics
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