# Free modified Rota-Baxter algebras and Hopf algebras

## TL;DR

This paper constructs free modified Rota-Baxter algebras and demonstrates how to equip them with bialgebra and Hopf algebra structures using cocycle methods, advancing algebraic theory.

## Contribution

It introduces a method to construct free modified Rota-Baxter algebras and establishes their bialgebra and Hopf algebra structures under certain conditions.

## Key findings

- Constructed free modified Rota-Baxter algebras.
- Established bialgebra structures via cocycle construction.
- Proved Hopf algebra structures for connected cases.

## Abstract

The notion of a modified Rota-Baxter algebra comes from the combination of those of a Rota-Baxter algebra and a modified Yang-Baxter equation. In this paper, we first construct free modified Rota-Baxter algebras. We then equip a free modified Rota-Baxter algebra with a bialgebra structure by a cocycle construction. Under the assumption that the generating algebra is a connected bialgebra, we further equip the free modified Rota-Baxter algebra with a Hopf algebra structure.