On the integration of transitive Lie algebroids

TL;DR

This paper provides a geometric understanding of the obstructions to integrating transitive Lie algebroids into Lie groupoids and offers explicit constructions when these obstructions are absent.

Contribution

It offers a geometric explanation of Crainic-Fernandes obstructions and constructs integrations for transitive Lie algebroids when obstructions vanish.

Findings

Crainic-Fernandes obstructions are geometrically explained.

Explicit integration methods are provided for cases with no obstructions.

Extension of approach to regular Lie algebroids is suggested.

Abstract

We revisit the problem of integrating Lie algebroids $A\Rightarrow M$ to Lie groupoids $G\rightrightarrows M$, for the special case that the Lie algebroid $A$ is transitive. We obtain a geometric explanation of the Crainic-Fernandes obstructions for this situation, and an explicit construction of the integration whenever these obstructions vanish. We also indicate an extension of this approach to regular Lie algebroids.