Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids
  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. 
Liqiang Cai, Jiefeng Liu and Yunhe Sheng
math.DG math-ph math.MP