On Hom-Lie Superalgebras

Abstract

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