On hom-Lie algebras

Abstract

In this paper, first we show that $(\g,[\cdot,\cdot],α)$ is a hom-Lie algebra if and only if $(Λ\g^*,α^*,d)$ is an $({\alpha }^{\ast },{\alpha }^{\ast })$-differential graded commutative algebra. Then, we revisit representations of hom-Lie algebras, and show that there are a series of coboundary operators. We also introduce the notion of an omni-hom-Lie algebra associated to a vector space and an invertible linear map. We show that regular hom-Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-hom-Lie algebra. The underlying algebraic structure of the omni-hom-Lie algebra is a hom-Leibniz algebra, or a hom-Lie 2-algebra.

15 pages

References in corpus (3)

• Hom-algebras and homology
• Hom-Lie 2-algebras
• The classical Hom-Yang-Baxter equation and Hom-Lie bialgebras