Cohomology and deformations of weighted Rota–Baxter operators

Abstract

Weighted Rota–Baxter operators on associative algebras are closely related to modified Yang–Baxter equations, splitting of algebras, and weighted infinitesimal bialgebras and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by λ-weighted relative Rota–Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal, and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation, which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota–Baxter operators in the Lie case and find a connection with the case of associative algebras.