On Hom–Lie algebras

Abstract

In this paper, first we show that is a Hom–Lie algebra if and only if is an 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.