Monomial Rota-Baxter operators of weight zero and averaging operators on the polynomial algebra

TL;DR

This paper classifies specific classes of monomial Rota-Baxter operators of weight zero on polynomial algebras and explores their relationship with monomial averaging operators, extending previous work on Rota-Baxter operators.

Contribution

It provides a classification of monomial Rota-Baxter operators of weight zero on bivariate polynomial algebras and establishes a connection with monomial averaging operators.

Findings

Classified non-increasing degree Rota-Baxter operators without kernel monomials.

Described operators mapping monomials to themselves with coefficients.

Demonstrated how to define monomial averaging operators from Rota-Baxter operators.

Abstract

Starting with the work S.H. Zheng, L. Guo and M. Rosenkranz (2015), Rota-Baxter operators are studied on the polynomial algebra. Injective Rota-Baxter operators of weight zero on $F[x]$ were described in 2021. We classify the following classes of monomial Rota-Baxter operators of weight zero on the polynomial algebra $F[x,y]$ and its augmentation ideal $F_0[x,y]$: 1) non-increasing in degree that do not contain monomials in the kernel, 2) mapping all monomials to themselves with a coefficient. In the context of these sets of operators, we show how one may define a monomial averaging operator by a given RB-operator and vice versa.