Twisted cohomology of Lie algebroids

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.