Monomial Rota-Baxter operators of nonzero weight on $F[x,y]$ coming from averaging operators

TL;DR

This paper classifies monomial Rota-Baxter operators of nonzero weight on two-variable polynomial algebras, focusing on those derived from averaging operators, extending prior work from one-variable cases.

Contribution

It provides a description of monomial Rota-Baxter operators of nonzero weight on $F[x,y]$ originating from averaging operators, a novel extension beyond previous one-variable studies.

Findings

Family of such operators explicitly described

Connection established between averaging operators and Rota-Baxter operators

Extension of classification from one-variable to two-variable case

Abstract

The intensive study of Rota-Baxter operators on the polynomial algebra $F[x]$ has been started with the work of S.H. Zheng, L. Guo, and M. Rosenkranz (2015). We deal with the case of two variables and monomial Rota-Baxter operators of nonzero weight. The family of such operators arisen from homomorphic averaging operators on $F[x,y]$ is described.