Linear-in-degree monomial Rota-Baxter of weight zero and averaging operators on $F[x, y]$ and $F_0[x, y]$

TL;DR

This paper classifies monomial Rota-Baxter operators of weight zero on bivariate polynomial algebras, linking them to monomial averaging operators, thereby advancing the understanding of algebraic operator structures.

Contribution

It provides a complete classification of monomial Rota-Baxter operators of weight zero on $F[x,y]$ derived from monomial averaging operators, extending prior work on polynomial algebras.

Findings

Classification of monomial Rota-Baxter operators of weight zero on $F[x,y]$

Connection established between Rota-Baxter and averaging operators

Extension of previous classifications to bivariate polynomial algebras

Abstract

Rota-Baxter operators on the polynomial algebra have been actively studied since the work of S.H. Zheng, L. Guo, and M. Rosenkranz (2015). Monomial operators of an arbitrary weight (2016), as well as injective operators of weight zero on $F[x]$ (2021), have been described. The author described monomial Rota-Baxter operators of nonzero weight on $F[x, y]$ coming from averaging operators (2023) and studied the connection between monomial Rota-Baxter operators and averaging operators (2024). The main result of the current work is the classification of monomial Rota-Baxter operators of weight zero on $F[x,y]$ coming from monomial linear-in-degree averaging operators.