Department of Mathematics, National Tsing Hua University

Abstract:
This talk explores the unifying geometric framework of quasi-symplectic groupoids introduced by Ping Xu. We begin by reviewing classical Hamiltonian G-spaces and Alekseev-Malkin-Meinrenken (AMM) quasi-Hamiltonian spaces, contrasting their momentum maps and reduction theorems. To reconcile these approaches, we introduce quasi-symplectic groupoids and their associated Hamiltonian $Gamma$-spaces. This generalized setting naturally accommodates both the closed symplectic forms of classical mechanics and the twisted structures of AMM theory. Finally, we present the generalized reduction theorem for Hamiltonian $Gamma$-spaces, demonstrating how it elegantly recovers both the classical Marsden-Weinstein reduction and the AMM reduction as distinct special cases.