Cohomology and deformations of twisted Rota-Baxter operators and NS-algebras

TL;DR

This paper develops a cohomology theory for twisted Rota-Baxter operators and NS-algebras, linking their deformations to $L_$-algebras and operads, and explores applications to algebraic structures like Reynolds operators.

Contribution

It introduces a cohomology framework for twisted Rota-Baxter operators and NS-algebras, connecting their deformations to $L_$-algebras and operadic methods, and applies this to Reynolds operators.

Findings

Constructed an $L_$-algebra for twisted Rota-Baxter operators.

Defined cohomology for NS-algebras via multiplicative operads.

Studied deformations of operators and algebras through the developed cohomology.

Abstract

The aim of this paper is twofold. In the first part, we consider twisted Rota-Baxter operators on associative algebras that were introduced by Uchino as a noncommutative analogue of twisted Poisson structures. We construct an $L_\infty$-algebra whose Maurer-Cartan elements are given by twisted Rota-Baxter operators. This leads to cohomology associated to a twisted Rota-Baxter operator. This cohomology can be seen as the Hochschild cohomology of a certain associative algebra with coefficients in a suitable bimodule. We study deformations of twisted Rota-Baxter operators by means of the above-defined cohomology. Application is given to Reynolds operators. In the second part, we consider NS-algebras of Leroux that are related to twisted Rota-Baxter operators in the same way dendriform algebras are related to Rota-Baxter operators. We define cohomology of NS-algebras using multiplicative operads and study their deformations in terms of the cohomology.