Crossed modules and the integrability of Lie brackets

TL;DR

This paper establishes a connection between the integrability of Lie brackets in Lie algebroids and the lifting obstructions of crossed modules of groupoids, providing a classification framework for extensions of transitive Lie groupoids.

Contribution

It reveals that the integrability obstruction of transitive Lie algebroids matches the lifting obstruction of associated crossed modules, extending to general extensions and linking to Lie groupoid classifications.

Findings

Integrability obstruction coincides with lifting obstruction of crossed modules.

Provides classification of extensions of transitive Lie groupoids.

Extends results to general extensions of integrable transitive Lie algebroids.

Abstract

We show that the integrability obstruction of a transitive Lie algebroid coincides with the lifting obstruction of a crossed module of groupoids associated naturally with the given algebroid. Then we extend this result to general extensions of integrable transitive Lie algebroids by Lie algebra bundles. Such a lifting obstruction is directly related with the classification of extensions of transitive Lie groupoids. We also give a classification of such extensions which differentiates to the classification of transitive Lie algebroids discussed in \cite{KCHM:new}.