On Hom-Lie Superalgebras

  • Baoling Guan
  • Liangyun Chen
  • Bing Sun


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.


Hom-Lie superalgebras Omni-hom-Lie superalgebras 

Mathematics Subject Classification

17B99 55U15 



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Authors and Affiliations

  1. 1.College of SciencesQiqihar UniversityQiqiharChina
  2. 2.School of Mathematics and StatisticsNortheast Normal UniversityChangchunChina

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