Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids

TL;DR

This paper extends the theory of Lie algebroids to the Hom-setting, introducing Hom-Lie algebroids, Hom-Lie bialgebroids, and Hom-Courant algebroids, and explores their properties and interrelations.

Contribution

It provides new definitions and structures for Hom-Lie algebroids and related concepts, generalizing classical Lie theory to the Hom-framework.

Findings

Hom-Lie algebroid structure on cotangent bundle of Hom-Poisson manifold

Base manifold of Hom-Lie bialgebroid is a Hom-Poisson manifold

Double of a Hom-Lie bialgebroid forms a Hom-Courant algebroid

Abstract

In this paper, first we modify the definition of a Hom-Lie algebroid introduced by Laurent-Gengoux and Teles and give its equivalent dual description. Many results that parallel to Lie algebroids are given. In particular, we give the notion of a Hom-Poisson manifold and show that there is a Hom-Lie algebroid structure on the pullback of the cotangent bundle of a Hom-Poisson manifold. Then we give the notion of a Hom-Lie bialgebroid, which is a natural generalization of a purely Hom-Lie bialgebra and a Lie bialgebroid. We show that the base manifold of a Hom-Lie bialgebroid is a Hom-Poisson manifold. Finally, we introduce the notion of a Hom-Courant algebroid and show that the double of a Hom-Lie bialgebroid is a Hom-Courant algebroid. The underlying algebraic structure of a Hom-Courant algebroid is a Hom-Leibniz algebra, or a Hom-Lie 2-algebra.