Rota-Baxter operators on involutive associative algebras

TL;DR

This paper develops a cohomology theory for Rota-Baxter operators on involutive associative algebras, linking it to Hochschild cohomology and involutive dendriform algebras, and explores its functorial properties.

Contribution

It introduces a cohomology framework for Rota-Baxter operators on involutive algebras and connects it with existing algebraic structures and functorial constructions.

Findings

Defined cohomology governing formal deformations of Rota-Baxter operators.

Linked the cohomology to Hochschild cohomology of involutive algebras.

Extended the Fard-Guo functor to the involutive case.

Abstract

In this paper, we consider Rota-Baxter operators on involutive associative algebras. We define cohomology for Rota-Baxter operators on involutive algebras that governs the formal deformation of the operator. This cohomology can be seen as the Hochschild cohomology of a certain involutive associative algebra with coefficients in a suitable involutive bimodule. We also relate this cohomology with the cohomology of involutive dendriform algebras. Finally, we show that the standard Fard-Guo construction of the functor from the category of dendriform algebras to the category of Rota-Baxter algebras restricts to the involutive case.