On hom-Lie algebras

TL;DR

This paper explores the structure of hom-Lie algebras, establishing their equivalence with certain differential graded algebras, and introduces omni-hom-Lie algebras to characterize regular hom-Lie algebra structures.

Contribution

It provides a new characterization of hom-Lie algebras via differential graded algebras and introduces omni-hom-Lie algebras for classifying regular hom-Lie structures.

Findings

Hom-Lie algebras are equivalent to specific differential graded algebras.

Introduction of omni-hom-Lie algebra associated with a vector space.

Regular hom-Lie algebra structures characterized by Dirac structures.

Abstract

In this paper, first we show that $(\g,[\cdot,\cdot],\alpha)$ is a hom-Lie algebra if and only if $(\Lambda \g^*,\alpha^*,d)$ is an $(\alpha^*,\alpha^*)$-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.