Deformations of algebroid stacks

TL;DR

This paper develops a deformation theory for algebroid stacks on etale groupoids by constructing a controlling DGLA, linking it to Hochschild cochains in the case of twisted function algebras.

Contribution

It introduces a DGLA framework for deformations of algebroid stacks and relates it to Hochschild cochains via characteristic classes of gerbes.

Findings

Constructed a DGLA controlling deformations of algebroid stacks.

Established a quasi-isomorphism between the DGLA and Hochschild cochains for twisted function algebroids.

Connected deformation theory with gerbe characteristic classes.

Abstract

In this paper we consider deformations of an algebroid stack on an etale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe.