Commutative modified Rota-Baxter algebras, shuffle products and Hopf algebras

TL;DR

This paper develops a systematic theory of commutative modified Rota-Baxter algebras, constructing free examples via shuffle products and endowing them with Hopf algebra structures using Hochschild cocycles.

Contribution

It introduces a new class of modified Rota-Baxter algebras, constructs free objects with explicit formulas, and establishes their Hopf algebra structure.

Findings

Construction of free commutative modified Rota-Baxter algebras

Explicit recursive and closed-form descriptions

Hopf algebra structure via Hochschild cocycle

Abstract

In this paper, we begin a systematic study of modified Rota-Baxter algebras, as an associative analogue of the modified classical Yang-Baxter equation. We construct free commutative modified Rota-Baxter algebras by a variation of the shuffle product and describe the structure both recursively and explicitly. We then provide these algebras with a Hopf algebra structure by applying a Hochschild cocycle.