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.
Categories: math.RA math.RT