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3-Hom-Lie Algebras Based on \(\sigma \)-Derivation and Involution

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Abstract

We show that, having a Hom-Lie algebra and an element of its dual vector space that satisfies certain conditions, one can construct a ternary totally skew-symmetric bracket and prove that this ternary bracket satisfies the Hom-Filippov-Jacobi identity, i.e. this ternary bracket determines the structure of 3-Hom-Lie algebra on the vector space of a Hom-Lie algebra. Then we apply this construction to two Hom-Lie algebras constructed on an associative, commutative algebra using \(\sigma \)-derivation and involution, and we obtain two 3-Hom-Lie algebras.

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Acknowledgements

The first author gratefully acknowledges that this work was partially supported by the institutional funding IUT20-57 of the Estonian Ministry of Education and Research and by the Doctoral School in Mathematics and Statistics of Estonia. The first author also thanks the program Nordplus Higher Education for financial support and the colleagues from the Division of Applied Mathematics of Mälardalen University for the creative atmosphere when working on this paper.

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Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of Tartu, J. Liivi 2, 50409, Tartu, Estonia

    Viktor Abramov

  2. Division of Applied Mathematics, Mälardalen University, Box 883, 721 23, Västerås, Sweden

    Sergei Silvestrov

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  1. Viktor Abramov
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  2. Sergei Silvestrov
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Correspondence to Viktor Abramov.

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Communicated by Michaela Vancliff

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Abramov, V., Silvestrov, S. 3-Hom-Lie Algebras Based on \(\sigma \)-Derivation and Involution. Adv. Appl. Clifford Algebras 30, 45 (2020). https://doi.org/10.1007/s00006-020-01068-6

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  • DOI: https://doi.org/10.1007/s00006-020-01068-6

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