Algebra, group, and Hopf Rota-Baxter operators

TL;DR

This paper explores the relationships between various types of Rota-Baxter operators on algebraic structures like groups, algebras, and Hopf algebras, focusing on the 4-dimensional Sweedler algebra.

Contribution

It investigates the connections between different Rota-Baxter operators on algebraic systems, specifically on the Sweedler algebra H_4, a non-cocommutative Hopf algebra.

Findings

Identified relationships between Rota-Baxter operators on H_4

Compared operators on groups, algebras, and Hopf algebras

Extended understanding of Rota-Baxter structures in non-cocommutative Hopf algebras

Abstract

We know definition of Rota--Baxter operators on different algebraic systems. For examples, on groups, on algebras, on Hopf algebras. On some algebraic systems it is possible to define different types of Rota--Baxter operators. For example, on group algebra it is possible to define Rota--Baxter operator as on associative algebra, group Rota--Baxter operator and Rota--Baxter operator as on a Hopf algebra. We are investigating the following question: What are connections between these operators? We are studying these questions for the Sweedler algebra $H_4$, that is 4-dimension non--cocommutative Hopf algebra.