On deformations of Lie algebroids

TL;DR

This paper explores the deformation cohomology of Lie algebroids by representing it as the cohomology of a specific subcomplex related to the 1-jet bundle, providing a new perspective on their deformations.

Contribution

It introduces a realization of Lie algebroid deformation cohomology as the cohomology of a subcomplex, extending previous definitions by Crainic and Moerdijk.

Findings

Cohomology of a subcomplex models deformation theory

Provides a new computational approach for Lie algebroid deformations

Links jet bundle structures with deformation cohomology

Abstract

For any Lie algebroid A, its 1-jet bundle JA is a Lie algebroid naturally and there is a representation \pi: JA ->DA. Denote by dJ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex (\Gamma(Hom(^\bulletJA,A)DA), dJ) of the cochain complex (\Gamma(Hom(^\bulletJA,A)), dJ).