Momentum Maps and Morita Equivalence

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 Γ ⇉ P . In particular, we study the reduction theory and prove that J(O)/Γ is a symplectic manifold for any Hamiltonian Γ-space (X J → P, ωX) (even though ωX ∈ Ω(X) may be degenerate), where O ⊂ P is a groupoid orbit. More generally, we prove that the intertwiner space (X1 ×P X2)/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 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.