Momentum Maps and Morita Equivalence

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 the ordinary Hamiltonian $G$-spaces, Lu's momentum maps of Poisson group actions, and group valued momentum maps of Alekseev--Malkin--Meinrenken. More precisely, we carry out the following program: (1) Define and study properties of quasi-symplectic groupoids; (2) Study the momentum map theory defined by a quasi-symplectic groupoid. In particular, we study the reduction theory and prove that the reduced space is always a symplectic manifold. More generally, we prove that the classical intertwiner space between two Hamiltonian $\Gamma$-spaces is always a symplectic manifold whenever it is a smooth manifold; (3) Study the Morita equivalence of quasi-symplectic groupoids. In particular, we prove that Morita equivalent quasi-symplectic groupoids give rise to equivalent momentum map theories and that the intertwiner space depends only on the Morita equivalence class. As a result, we recover various well-known results concerning equivalence of momentum maps including Alekseev-- Ginzburg--Weinstein linearization theorem and Alekseev--Malkin--Meinrenken equivalence theorem between quasi-Hamiltonian spaces and Hamiltonian loop group sapces.