Twisted Rota-Baxter operators on Hom-Lie algebras

Abstract

<abstract><p>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.</p></abstract>

Generalized Reynolds Operators on Hom-Lie Triple Systems

“…Das also developed the notions of generalized Reynolds operators on Lie algebras and NS-Lie algebras in [24]. Generalized Reynolds operators on other algebraic structures have also been widely studied, including 3-Lie algebras [25,26], 3-Hom-Lie algebras [27], Hom-Lie algebras [28], Lie-Yamaguti algebras [29], Lie triple systems [29,30] and Lie supertriple systems [31].…”