Lie algebroid modules and representations up to homotopy

TL;DR

This paper explores the connection between Lie algebroid modules and representations up to homotopy, establishing a correspondence that links these two generalizations of Lie algebroid representations.

Contribution

It demonstrates a noncanonical method to derive a representation up to homotopy from a Lie algebroid module and proves their equivalence up to natural isomorphism.

Findings

Established a one-to-one correspondence between Lie algebroid modules and representations up to homotopy.

Showed that the derived representations up to homotopy are equivalent in a natural sense.

Provided a framework connecting two different generalizations of Lie algebroid representations.

Abstract

We establish a relationship between two different generalizations of Lie algebroid representations: representation up to homotopy and Vaintrob's Lie algebroid modules. Specifically, we show that there is a noncanonical way to obtain a representation up to homotopy from a given Lie algebroid module, and that any two representations up to homotopy obtained in this way are equivalent in a natural sense. We therefore obtain a one-to-one correspondence, up to equivalence.