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Momentum maps and Morita equivalence. (English) Zbl 1106.53057

A theory of Hamiltonian actions of quasi-symplectic groupoids is presented, in order to unify different momentum map theories such as: Hamiltonian \(G\)-spaces [J. Marsden and A. Weinstein, Rep. Math. Phys. 5, 121–130 (1974; Zbl 0327.58005)], Lu-Weinstein’s momentum maps of Poisson group actions [J.-H. Lu and A. Weinstein, J. Differ. Geom. 31, No. 2, 501–526 (1990; Zbl 0673.58018)], and the group-values momentum maps of A. Alekseev, A. Malkin and E. Meinrenken [J. Differ. Geom. 48, No. 3, 445–495 (1998; Zbl 0948.53045)].
First, the author defines and studies quasi-symplectic groupoids. Second, he introduces the notion of Hamiltonian spaces for quasi-symplectic groupoids (giving some examples), and the corresponding reduction theorem. Finally, he studies the Morita equivalence, proving that Morita equivalent quasi-symplectic groupoids define equivalent momentum map theories. As a consequence, several results concerning the equivalence of the above-mentioned momentum map theories are recovered.

MSC:

53D20 Momentum maps; symplectic reduction
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