Representations up to homotopy of Lie algebroids

TL;DR

This paper introduces the concept of representations up to homotopy for Lie algebroids, explores their applications in defining the adjoint representation, and connects the Weil algebra to equivariant cohomology models.

Contribution

It develops the theory of representations up to homotopy for Lie algebroids and links the Weil algebra to Kalkman's BRST model for equivariant cohomology.

Findings

Representation up to homotopy defines the adjoint representation.

Lie algebroid cohomology controls structure deformations.

Weil algebra matches Kalkman's BRST model for equivariant cohomology.

Abstract

We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid is defined and shown to coincide with Kalkman's BRST model for equivariant cohomology in the case of group actions.