Rota-Baxter operators on the polynomial algebras, integration and averaging operators

TL;DR

This paper explores the connection between Rota-Baxter operators and integration in polynomial algebras, classifying monomial and injective operators, and applying averaging and double product techniques.

Contribution

It introduces a classification of Rota-Baxter operators on polynomial algebras, linking them to averaging operators and double product structures, advancing algebraic understanding.

Findings

Classified monomial Rota-Baxter operators using averaging operators.

Analyzed injective Rota-Baxter operators via double product structures.

Provided new algebraic characterizations of Rota-Baxter operators on polynomial algebras.

Abstract

Rota-Baxter operators are an algebraic abstraction of integration. Following this classical connection, we study the relationship between Rota-Baxter operators and integrals in the case of the polynomial algebra $\mathbf{k}[x]$. We consider two classes of Rota-Baxter operators, monomial ones and injective ones. For the first class, we apply averaging operators to determine monomial Rota-Baxter operators. For the second class, we make use of the double product on Rota-Baxter algebras.