Lie Groupoid Deformations and Convolution Algebras

TL;DR

This paper establishes a connection between geometric deformations of Lie groupoids and algebraic deformations of their convolution algebras, revealing how classical limits relate to quantization in the context of Lie algebroids.

Contribution

It introduces a morphism linking the deformation complex of a Lie groupoid to the Hochschild complex of its convolution algebra, and demonstrates how classical limits induce the van Est map in deformation cohomology.

Findings

Mapped geometric deformations to Hochschild cohomology classes.

Showed the van Est map arises from classical limits of quantization.

Connected deformation theory of Lie groupoids with algebraic structures.

Abstract

We define a morphism from the deformation complex of a Lie groupoid to the Hochschild complex of its convolution algebra, and show that it maps the class of a geometric deformation to the algebraic class of the induced deformation in Hochschild cohomology. Applied to the adiabatic groupoid, we show that the van Est map to deformation cohomology of Lie algebroids is induced by taking the classical limit of a quantization map on the dual of the Lie algebroid.