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

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.

25 pages

References in corpus (3)

• Notes on Formal Deformations of Hom-associative and Hom-Lie Algebras
• Twisting Poisson algebras, coPoisson algebras and Quantization
• Left-symmetric algebroids

Cited by in corpus (8)

• On Hom-Gerstenhaber algebras and Hom-Lie algebroids
• Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation
• Cohomology and deformations of compatible Hom-Lie algebras
• A complement on representations of Hom-Lie algebras
• On split regular Hom-Leibniz-Rinehart algebras
• Kähler-Norden structures on Hom-Lie group and Hom-Lie algebras
• Equivalent description of Hom-Lie algebroids
• Hom-Lie groups of a class of Hom-Lie algebra