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(𝒪)/γ is a symplectic manifold for any Hamiltonian γ-space (X →J P, ωX) (even though ωX ∈ ω2(X) may be degenerate), where 𝒪 ⊂ 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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Geometry and Topology