On the homotopy invariance of the twisted Lie algebroid cohomology

TL;DR

This paper proves that twisted Lie algebroid cohomology remains invariant under homotopies, extends classical results like the Poincaré lemma, and provides tools for explicit computation of these cohomologies.

Contribution

It establishes homotopy invariance for twisted Lie algebroid cohomology and develops a generalized Poincaré lemma and Künneth formula for these structures.

Findings

Homotopy invariance of twisted Lie algebroid cohomology

Generalized Poincaré lemma in the transitive case

Künneth formula for Lie algebroid cohomologies

Abstract

Twisted Lie algebroid cohomologies, i.e. with values in representations, are shown to be Lie algebroid homotopy-invariant. Several important classes of examples are discussed. As an application, a generalized version of the Poincar\'e lemma is given in the transitive case. Together with the Mayer-Vietoris theorem, which holds in this more general context as well, this leads to a K\"unneth formula for the cohomologies of Lie algebroids with values in representations. In particular, this comprehensive paper gives a systematic way to compute explicitly the twisted Lie algebroid cohomologies of some transitive and almost transitive Lie algebroids.