The cohomology and deformations of O-operators on BiHom-associative algebras

Abstract

We first generalize the cohomology of $O$-operators on BiHom-associative algebras by construct a graded Lie-algebra, in which the Maurer-Cartan elements are characterized by the given $O$-operator, and show that the cohomology represents the Hochschild cohomology of a certain BiHom-associative algebra with coefficients in a bimodule. Next, we study the linear and formal deformations of $O$-operators on BiHom-associative algebras, which are controlled by the Hochschild cohomology. Finally, as applications, we introduce the deformations of BiHom-associative r-matrices and infinitesimal BiHom-bialgebras on certain regular BiHom-associative algebras.