Cohomology of nonabelian embedding tensors on Hom-Lie algebras

Wen Teng; Jiulin Jin; Yu Zhang; Wen Teng; Jiulin Jin; Yu Zhang

doi:10.3934/math.20231079

AIMS Mathematics

2023, Volume 8, Issue 9: 21176-21190. doi: 10.3934/math.20231079

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Research article

Cohomology of nonabelian embedding tensors on Hom-Lie algebras

1.
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
2.
College of Mathematics and Information Science, Guiyang University, Guiyang 550005, China

Received: 07 May 2023 Revised: 16 June 2023 Accepted: 20 June 2023 Published: 03 July 2023
MSC : 17A32, 17B38, 17B56, 17B61

In this paper, we generalize known results of nonabelian embedding tensor to the Hom setting. We introduce the concept of Hom-Leibniz-Lie algebra, which is the basic algebraic structure of nonabelian embedded tensors on Hom-Lie algebras and can also be regarded as a nonabelian generalization of Hom-Leibniz algebra. Moreover, we define a cohomology of nonabelian embedding tensors on Hom-Lie algebras with coefficients in a suitable representation. The first cohomology group is used to describe infinitesimal deformations as an application. In addition, Nijenhuis elements are used to describe trivial infinitesimal deformations.
- Hom-Lie algebra,
- Hom-Leibniz-Lie algebra,
- nonabelian embedding tensor,
- cohomology,
- deformation
Citation: Wen Teng, Jiulin Jin, Yu Zhang. Cohomology of nonabelian embedding tensors on Hom-Lie algebras[J]. AIMS Mathematics, 2023, 8(9): 21176-21190. doi: 10.3934/math.20231079

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Abstract

In this paper, we generalize known results of nonabelian embedding tensor to the Hom setting. We introduce the concept of Hom-Leibniz-Lie algebra, which is the basic algebraic structure of nonabelian embedded tensors on Hom-Lie algebras and can also be regarded as a nonabelian generalization of Hom-Leibniz algebra. Moreover, we define a cohomology of nonabelian embedding tensors on Hom-Lie algebras with coefficients in a suitable representation. The first cohomology group is used to describe infinitesimal deformations as an application. In addition, Nijenhuis elements are used to describe trivial infinitesimal deformations.

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