Automorphisms of Lie groupoids and symplectic reduction on orbifolds

TL;DR

This paper develops a framework for automorphisms of Lie groupoids and applies it to Hamiltonian actions on orbifolds, demonstrating that symplectic reduction yields orbifolds or symplectic Lie 2-groupoids.

Contribution

It introduces the 2-group BAut(X) for Lie groupoid automorphisms and links 2-group homomorphisms to Kan fibrations, advancing the understanding of symplectic reduction on orbifolds.

Findings

Symplectic reduction of Hamiltonian actions results in symplectic Lie 2-groupoids.

Under certain conditions, the reduction remains an orbifold.

A slice theorem for group actions on Lie groupoids is established.

Abstract

In this paper, the 2-group BAut(X) of automorphisms of a Lie groupoid X is constructed. Considering the 2-group G action on X, we explain the equivalence between 2-group homomorphisms from G to BAut(X) with Kan fibrations over G with fiber X. This justifies the notion of Kan fibration for 2-group actions on Lie groupoids. As an application, we formulate Hamiltonian actions of \'etale Lie 2-groups on orbifolds in terms of Kan fibrations and study the symplectic reductions. We show that, in general, the reduction is in fact a symplectic Lie 2-groupoid, and under certain isotropic free condition, the reduction is still an orbifold. Also the slice theorem of a group G action on Lie groupoids is proved.