Twisted Rota-Baxter operators and Reynolds operators on Lie algebras and NS-Lie algebras

Abstract

In this paper, we introduce twisted Rota-Baxter operators on Lie algebras as an operator analogue of twisted r-matrices. We construct a suitable $L_{\infty }$-algebra whose Maurer-Cartan elements are given by twisted Rota-Baxter operators. This allows us to define cohomology of a twisted Rota-Baxter operator. This cohomology can be seen as the Chevalley-Eilenberg cohomology of a certain Lie algebra with coefficients in a suitable representation. We study deformations of twisted Rota-Baxter operators from cohomological points of view. Some applications are given to Reynolds operators and twisted r-matrices. Next, we introduce a new algebraic structure, called NS-Lie algebras, that are related to twisted Rota-Baxter operators in the same way pre-Lie algebras are related to Rota-Baxter operators. We end this paper by considering twisted generalized complex structures on modules over Lie algebras.

arXiv commentTitle has been changed slightly and some minor revision has been done. (Final version)

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Cited by in corpus (3)

• Cohomology and deformations of weighted Rota-Baxter operators
• Deformations and homotopy theory of Rota-Baxter algebras of any weight
• Reynolds $n$-Lie algebras and NS-$n$-Lie algebras