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

TL;DR

This paper introduces twisted Rota-Baxter operators on Lie algebras, develops their cohomology and deformation theory, and explores related structures like NS-Lie algebras and twisted generalized complex structures.

Contribution

It defines twisted Rota-Baxter operators, constructs their cohomology via $L_$-algebras, and introduces NS-Lie algebras, extending the theory of Rota-Baxter operators.

Findings

Cohomology of twisted Rota-Baxter operators is developed.

Deformation theory for these operators is established.

Applications to Reynolds operators and twisted r-matrices are demonstrated.

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.