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- W1592068265 abstract "We 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 ordinary Hamiltonian $G$-spaces, Lu's momentum maps of Poisson group actions, and the group-valued momentum maps of Alekseev–Malkin–Meinrenken. More precisely, we carry out the following program: (1) We define and study properties of quasi-symplectic groupoids. (2) We study the momentum map theory defined by a quasi-symplectic groupoid $Gamma rightrightarrows P$. In particular, we study the reduction theory and prove that $J^{-1} mathcal {O})/ Gamma$ is a symplectic manifold for any Hamiltonian $Gamma$-space $(X stackrel {J} rightarrow P, omega_X$ (even though $omega_X in Omega^2 (X)$ may be degenerate), where $mathcal{O} in P$ is a groupoid orbit. More generally, we prove that the intertwiner space $(X_1 times_{P} overline{X_2}$ between two Hamiltonian $Gamma$-spaces $X_1$ and $X_2$ is a symplectic manifold (whenever it is a smooth manifold). (3) We study Morita equivalence of quasi-symplectic groupoids. In particular, we prove that Morita equivalent quasi-symplectic groupoids give rise to equivalent momentum map theories. Moreover the intertwiner space $(X_1 times_P overline{X_2}$depends only on the Morita equivalence class. As a result, we recover various well-known results concerning equivalence of momentum maps including the Alekseev–Ginzburg–Weinstein linearization theorem and the Alekseev–Malkin–Meinrenken equivalence theorem between quasi-Hamiltonian spaces and Hamiltonian loop group spaces." @default.
- W1592068265 created "2016-06-24" @default.
- W1592068265 creator A5069719518 @default.
- W1592068265 date "2004-06-01" @default.
- W1592068265 modified "2023-09-30" @default.
- W1592068265 title "Momentum Maps and Morita Equivalence" @default.
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- W1592068265 doi "https://doi.org/10.4310/jdg/1102536203" @default.
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