# Twisted cohomology of Lie algebroids

## TL;DR

This paper introduces a new twisted cohomology for Lie algebroids, depending on an odd cocycle, which generalizes several existing cohomology theories and depends only on its cohomology class.

## Contribution

It defines the twisted cohomology for Lie algebroids and proves its dependence solely on the cohomology class of the twisting cocycle, unifying various known theories.

## Key findings

- The twisted cohomology depends only on the cohomology class of the cocycle.
- Examples show the new cohomology encompasses known theories.
- The construction generalizes existing cohomology concepts.

## Abstract

In this short note we define a new cohomology for a Lie algebroid $\mathcal{A}$, that we call the \emph{twisted cohomology} of $\mathcal{A}$ by an odd cocycle $\theta$ in the Lie algebroid cohomology of $\mathcal{A}$. We proof that this cohomology only depends on the Lie algebroid cohomology class $[\theta]$ of the odd cocycle $\theta$. We give a few examples showing that this new cohomology encompasses various well-known cohomology theories.