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In this paper, we introduce twisted Rota-Baxter operators on Lie algebras asan operator analogue of twisted r-matrices. We construct a suitable$L_\infty$-algebra whose Maurer-Cartan elements are given by twistedRota-Baxter operators. This allows us to define cohomology of a twistedRota-Baxter operator. This cohomology can be seen as the Chevalley-Eilenbergcohomology of a certain Lie algebra with coefficients in a suitablerepresentation. We study deformations of twisted Rota-Baxter operators fromcohomological points of view. Some applications are given to Reynolds operatorsand twisted r-matrices. Next, we introduce a new algebraic structure, calledNS-Lie algebras, that are related to twisted Rota-Baxter operators in the sameway pre-Lie algebras are related to Rota-Baxter operators. We end this paper byconsidering twisted generalized complex structures on modules over Liealgebras.

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