On hom-Lie algebras

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 $(α^*,α^*)$-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.

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