Twisted Rota-Baxter operators on Hom-Lie algebras

Senrong Xu; Wei Wang; Jia Zhao; Senrong Xu; Wei Wang; Jia Zhao

doi:10.3934/math.2024129

AIMS Mathematics

2024, Volume 9, Issue 2: 2619-2640. doi: 10.3934/math.2024129

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Research article

Twisted Rota-Baxter operators on Hom-Lie algebras

1.
School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu 212013, China
2.
Institute of Applied System Analysis, Jiangsu University, Zhenjiang, Jiangsu 212013, China
3.
School of Sciences, Nantong University, Nantong, Jiangsu 226019, China

Received: 09 October 2023 Revised: 19 December 2023 Accepted: 21 December 2023 Published: 26 December 2023
MSC : 17B38, 17B61, 17B56

Uchino initiated the investigation of twisted Rota-Baxter operators on associative algebras. Relevant studies have been extensive in recent times. In this paper, we introduce the notion of a twisted Rota-Baxter operator on a Hom-Lie algebra. By utilizing higher derived brackets, we establish an explicit $ L_{\infty} $-algebra whose Maurer-Cartan elements are precisely twisted Rota-Baxter operators on Hom-Lie algebra s. Additionally, we employ Getzler's technique of twisting $ L_\infty $-algebras to establish the cohomology of twisted Rota-Baxter operators. We demonstrate that this cohomology can be regarded as the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation. Finally, we study the linear and formal deformations of twisted Rota-Baxter operators by using the cohomology defined above. We also show that the rigidity of a twisted Rota-Baxter operator can be derived from Nijenhuis elements associated with a Hom-Lie algebra.
- twisted Rota-Baxter operators,
- Hom-Lie algebra s,
- Maurer-Cartan elements,
- $ L_\infty $-algebras,
- cohomologies,
- deformations
Citation: Senrong Xu, Wei Wang, Jia Zhao. Twisted Rota-Baxter operators on Hom-Lie algebras[J]. AIMS Mathematics, 2024, 9(2): 2619-2640. doi: 10.3934/math.2024129

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Abstract

Uchino initiated the investigation of twisted Rota-Baxter operators on associative algebras. Relevant studies have been extensive in recent times. In this paper, we introduce the notion of a twisted Rota-Baxter operator on a Hom-Lie algebra. By utilizing higher derived brackets, we establish an explicit $ L_{\infty} $-algebra whose Maurer-Cartan elements are precisely twisted Rota-Baxter operators on Hom-Lie algebra s. Additionally, we employ Getzler's technique of twisting $ L_\infty $-algebras to establish the cohomology of twisted Rota-Baxter operators. We demonstrate that this cohomology can be regarded as the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation. Finally, we study the linear and formal deformations of twisted Rota-Baxter operators by using the cohomology defined above. We also show that the rigidity of a twisted Rota-Baxter operator can be derived from Nijenhuis elements associated with a Hom-Lie algebra.

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