Deformations of Lie groupoids

TL;DR

This paper investigates the deformation theory of Lie groupoids using cohomology, establishing fundamental properties and applications such as rigidity and normal form results.

Contribution

It introduces a cohomology framework for Lie groupoid deformations, proving Morita invariance, a van Est theorem, and vanishing results in the proper case.

Findings

Deformation cohomology provides an intrinsic model for Lie groupoid cohomology.

Morita invariance of the deformation cohomology.

Vanishing theorem for proper Lie groupoids.

Abstract

We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Combined with Moser's deformation arguments for groupoids, we obtain several rigidity and normal form results.