### 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 quasisymplectic groupoid γ ⇒ P. In particular, we study the reduction theory and prove that J^{−1}(풪)/γ is a symplectic manifold for any Hamiltonian γ-space (X →^{J} P, ω_{X}) (even though ω_{X} ∈ ω^{2}(X) may be degenerate), where 풪 ⊂ P is a groupoid orbit. More generally, we prove that the intertwiner space (X_{1}×P X^{¯} _{2})/γ between two Hamiltonian γ-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} ×P X_{2})/γ depends only on the Morita equivalence class. As a result, we recover various wellknown 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.

Original language | English (US) |
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Pages (from-to) | 289-333 |

Number of pages | 45 |

Journal | Journal of Differential Geometry |

Volume | 67 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2004 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology

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## Cite this

*Journal of Differential Geometry*,

*67*(2), 289-333. https://doi.org/10.4310/jdg/1102536203