Representation up to homotopy of Hom-Lie algebroids

TL;DR

This paper extends the concept of representations up to homotopy to hom-Lie algebroids, establishing a framework that relates these representations to extensions of the structures.

Contribution

It introduces a novel approach to hom-Lie algebroids by applying representations up to homotopy, connecting them to their extensions.

Findings

Constructed a representation up to homotopy for hom-Lie algebroids

Linked length 1 representations to extensions of hom-Lie algebroids

Provided a new perspective on the structure of hom-Lie algebroids

Abstract

A hom-Lie algebroid is a vector bundle together with a Lie algebroid like structure which is twisted by a homomorphism. In this paper we use the idea of representations up to homotopy of Lie algebroids to construct a same structure for hom-Lie algebroids and we will explain how representations up to homotopy of length 1 are related to extensions of hom-Lie algebroids.