Extensions of Lie Brackets

TL;DR

This paper develops a comprehensive framework for extending Lie algebroids using Ehresmann connections, exploring their cohomological properties, integrability conditions, and groupoid integrations, thus broadening the understanding of Lie algebroid extensions.

Contribution

It introduces a general approach to Lie algebroid extensions, including non-abelian cases and those over different bases, with new insights into their cohomology and integrability.

Findings

Established a filtration in cohomology for Lie algebroid extensions.

Described the integration of extensions via groupoids with fixed connections.

Analyzed the conditions for integrability of Lie algebroid extensions.

Abstract

We provide a framework for extensions of Lie algebroids, including non-abelian extensions and Lie algebroids over different bases. Our approach involves Ehresmann connections, which allows straight generalizations of classical constructions. We exhibit a filtration in cohomology and explain the associated spectral sequence. We also give a description of the groupoid integrating an extension in case a complete connection can be fixed. The problem of integrability is also studied.