Extension of algebroids Part I: The Construction
Simon-Raphael Fischer

TL;DR
This paper introduces a canonical construction for extending Lie algebroids using covariant adjustments and Cartan connections, generalizing Maurer-Cartan forms with curvature considerations.
Contribution
It develops a new framework for extending Lie algebroids via curvature-inclusive constructions, including a Cartan connection approach and obstruction theory.
Findings
Constructed a canonical extension method for Lie algebroids using covariant adjustments.
Linked Cartan connections with obstruction theory for Lie algebroid extensions.
Provided examples in Poisson geometry and crossed modules of Lie groupoids.
Abstract
In this series of two papers we will generalise the concept of extending a Lie algebroid by a Lie algebra bundle, leading to a notion of extending a Lie algebroid by another Lie algebroid whose orbits lie in the orbits of the former algebroid. The resulting Lie algebroid's anchor will be the sum of the two initial anchors such that the constructions will be similar to matched pairs of Lie algebroids, but with the key difference that we will allow a curvature. In this part of this series we will focus on the canonical construction making use of strict covariant adjustments, a generalisation of Maurer-Cartan forms in the context of gauge theories equipped with a Lie groupoid action instead of a Lie group action. That is, a Cartan connection with certain conditions on the curvature. The second paper will introduce and explain the obstruction of the extension provided here. Examples will…
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