Extensions of Lie Brackets
Olivier Brahic

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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Nonlinear Waves and Solitons · Advanced Topics in Algebra
