### 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 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Journal of Differential Geometry*,

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

}

*Journal of Differential Geometry*, vol. 67, no. 2, pp. 289-333. https://doi.org/10.4310/jdg/1102536203

**Momentum maps and morita equivalence.** / Xu, Ping.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Momentum maps and morita equivalence

AU - Xu, Ping

PY - 2004/1/1

Y1 - 2004/1/1

N2 - 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, ωX) (even though ωX ∈ ω2(X) may be degenerate), where 풪 ⊂ P is a groupoid orbit. More generally, we prove that the intertwiner space (X1×P X¯ 2)/γ between two Hamiltonian γ-spaces X1 and X2 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 (X1 ×P X2)/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.

AB - 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, ωX) (even though ωX ∈ ω2(X) may be degenerate), where 풪 ⊂ P is a groupoid orbit. More generally, we prove that the intertwiner space (X1×P X¯ 2)/γ between two Hamiltonian γ-spaces X1 and X2 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 (X1 ×P X2)/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.

UR - http://www.scopus.com/inward/record.url?scp=28844461279&partnerID=8YFLogxK

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U2 - 10.4310/jdg/1102536203

DO - 10.4310/jdg/1102536203

M3 - Article

VL - 67

SP - 289

EP - 333

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 2

ER -