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 analog of twisted r-matrices. We construct a suitable L∞-algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This allows us to define the 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 is 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.