Momentum Maps and Morita Equivalence

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dc.creatorXu, Ping
dc.date2003-07-24
dc.date2004-07-26
dc.date.accessioned2026-07-07T04:59:52Z
dc.date.available2026-07-07T04:59:52Z
dc.descriptionWe introduce quasi-symplectic groupoids and explain their relation with momentum map theories. This approach enables us to unify into a single framework various momentum map theories, including the ordinary Hamiltonian $G$-spaces, Lu's momentum maps of Poisson group actions, and group valued momentum maps of Alekseev--Malkin--Meinrenken. More precisely, we carry out the following program: (1) Define and study properties of quasi-symplectic groupoids; (2) Study the momentum map theory defined by a quasi-symplectic groupoid. In particular, we study the reduction theory and prove that the reduced space is always a symplectic manifold. More generally, we prove that the classical intertwiner space between two Hamiltonian $Γ$-spaces is always a symplectic manifold whenever it is a smooth manifold; (3) Study the Morita equivalence of quasi-symplectic groupoids. In particular, we prove that Morita equivalent quasi-symplectic groupoids give rise to equivalent momentum map theories and that the intertwiner space depends only on the Morita equivalence class. As a result, we recover various well-known results concerning equivalence of momentum maps including Alekseev-- Ginzburg--Weinstein linearization theorem and Alekseev--Malkin--Meinrenken equivalence theorem between quasi-Hamiltonian spaces and Hamiltonian loop group sapces.
dc.description34 pages, Latex file, typos corrected, final version to appear in J. Diff. Geom
dc.identifierhttps://arxiv.org/abs/math/0307319
dc.identifierhttp://arxiv.org/abs/math/0307319
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/68161
dc.subjectSymplectic Geometry
dc.subjectMathematical Physics
dc.subject58F05
dc.titleMomentum Maps and Morita Equivalence
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