Atiyah classes of Lie algebroid homotopy modules

TL;DR

This paper introduces a new cohomology class called the Atiyah class for Lie algebroid homotopy modules, revealing its independence from extensions and its relation to classical Atiyah classes.

Contribution

It constructs a novel Atiyah class for Lie algebroid homotopy modules and establishes its invariance and connection to classical Atiyah classes through quasi-isomorphisms.

Findings

Existence of a cohomology class independent of extensions.

The Atiyah class vanishes when a homotopy-compatible extension exists.

Relation between the new Atiyah class and classical Atiyah classes via quasi-isomorphisms.

Abstract

For a Lie algebroid pair $A\hookrightarrow L$ we study cocycles constructed from the extension to $L$ of the higher connection forms of a representation up to homotopy $E$ of the Lie algebroid $A$. We show that there exists a cohomology class with values in the endomorphism bundle of $E$ that is independent of the extension above and vanishes whenever a homotopy $A$-compatible extension exists. Whenever the representation up to homotopy $E$ is the resolution of a Lie algebroid representation $K$ of $A$, it is shown that there exists a quasi-isomorphism sending the new Atiyah class to the classical one, associated to extensions to $L$ of the Lie algebroid representation $K$.