Deformations of associative Rota-Baxter operators

Abstract

Rota-Baxter operators and more generally 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 O-operators. This allows us to construct cohomology for an O-operator. This cohomology can also be seen as the Hochschild cohomology of a certain algebra with coefficients in a suitable bimodule. Next, we study linear and formal deformations of an O-operator which are governed by the above-defined cohomology. We introduce Nijenhuis elements associated with an O-operator which give rise to trivial deformations. As applications, we conclude deformations of weight zero Rota-Baxter operators, associative r-matrices and averaging operators.