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.
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Communicated by Michaela Vancliff
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Supported by NNSF of China (nos. 11171055 and 11471090), NSF of Jilin Province (no. 20170101048 JC), Special Project of Basic Business for Heilongjiang Provincial Education Department (no.135209255) and Postdoctoral Scientific Research Developmental Fund of Heilongjiang (no. LBH-Q17175).
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Guan, B., Chen, L. & Sun, B. On Hom-Lie Superalgebras. Adv. Appl. Clifford Algebras 29, 16 (2019). https://doi.org/10.1007/s00006-018-0932-1
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DOI: https://doi.org/10.1007/s00006-018-0932-1