Deformations of associative Rota-Baxter operators

TL;DR

This paper develops a cohomology theory for associative Rota-Baxter operators using graded Lie algebras and explores their deformations, including trivial ones via Nijenhuis elements, with applications to Rota-Baxter operators and associative r-matrices.

Contribution

It constructs an explicit graded Lie algebra framework for $ ext{O}$-operators and introduces a cohomology theory to study their deformations, including trivial deformations via Nijenhuis elements.

Findings

Constructed a graded Lie algebra with Maurer-Cartan elements as $ ext{O}$-operators.

Developed a cohomology theory for $ ext{O}$-operators related to Hochschild cohomology.

Analyzed linear and formal deformations, identifying Nijenhuis elements as trivial deformations.

Abstract

Rota-Baxter operators and more generally $\mathcal{O}$-operators on associative algebras are important in probability, combinatorics, associative Yang-Baxter equation and splitting of algebras. Using a method of Uchino, we construct an explicit graded Lie algebra whose Maurer-Cartan elements are given by $\mathcal{O}$-operators. This allows us to construct a cohomology for an $\mathcal{O}$-operator. This cohomology can also be seen as the Hochschild cohomology of a certain algebra with coefficients in a suitable representation. Next, we study linear and formal deformations of an $\mathcal{O}$-operator which are governed by the above-defined cohomology. We introduce Nijenhuis elements associated with an $\mathcal{O}$-operator which give rise to trivial deformations. As an application, we conclude deformations of weight zero Rota-Baxter operators and associative {\bf r}-matrices.