Cohomology and deformations of weighted Rota-Baxter operators

TL;DR

This paper develops a cohomology theory for weighted Rota-Baxter operators on associative algebras, linking it to deformation theory, Hochschild cohomology, and the Lie algebra case, with implications for mathematical physics.

Contribution

It constructs a differential graded Lie algebra framework for weighted Rota-Baxter operators and introduces cohomology and deformation theories, including Nijenhuis elements and obstruction classes.

Findings

Defined cohomology for weighted Rota-Baxter operators.

Analyzed linear, formal, and finite order deformations cohomologically.

Connected associative and Lie algebra cases of the cohomology theory.

Abstract

Weighted Rota-Baxter operators on associative algebras are closely related to modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal bialgebras, and play an important role in mathematical physics. For any $\lambda \in {\bf k}$, we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by $\lambda$-weighted relative Rota-Baxter operators. Using such characterization, we define the cohomology of a $\lambda$-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 $\lambda$-weighted relative Rota-Baxter operators in the Lie case and find a connection with the case of associative algebras.