Monomial Rota-Baxter operators on free commutative non-unital algebra

TL;DR

This paper classifies monomial Rota-Baxter operators on polynomial algebras without constant terms, providing a comprehensive understanding of their structure in one and multiple variables.

Contribution

It offers a complete classification of monomial Rota-Baxter operators on polynomial algebras, including injective operators in multiple variables, advancing the algebraic theory of Rota-Baxter operators.

Findings

Classified monomial Rota-Baxter operators on one-variable polynomial algebra.

Described injective monomial Rota-Baxter operators in multiple variables.

Provided structural insights into Rota-Baxter operators without constant terms.

Abstract

A Rota-Baxter operator defined on the polynomial algebra is called monomial if it maps each monomial to a monomial with some coefficient. We classify monomial Rota-Baxter operators defined on the algebra of polynomials in one variable without constant term. We also describe injective monomial Rota--Baxter operators of nonzero weight on the algebra of polynomials in several variables without constant term.