Representation up to Homotopy and Hom-Lie Algebroid Modules

Abstract

In this paper we introduce the concept of hom-Lie algebroid modules and hom-Lie algebroids. Then we show the correspondence between hom-Lie algebroid modules and representation up to homotopy of hom-Lie algebroids. Because of the effective role of representation theory and Lie algebraic structures in particle physics, we show the correspondence between bi-graded hom-Lie algebraic modules and hom-Lie algebraist. At the end, we study some properties of representation up to homotopy, using the language of hom-Lie algebroid modules.