Rota-Baxter operators of weight zero on the matrix algebra of order three without unit in kernel

TL;DR

This paper classifies Rota-Baxter operators of weight zero on 3x3 matrix algebras over certain fields, focusing on those with non-zero image of the identity, contributing to the understanding of solutions to the associative Yang-Baxter equation.

Contribution

It provides a partial classification of Rota-Baxter operators on M_3(F) with specific properties, using computational algebra tools.

Findings

Identified all Rota-Baxter operators of weight zero with R(1) ≠ 0 on M_3(F).

Contributed to the classification of solutions to the associative Yang-Baxter equation.

Utilized computer algebra system Singular for computations.

Abstract

We describe all Rota-Baxter operators $R$ of weight zero on the matrix algebra $M_3(F)$ over a quadratically closed field $F$ of characteristic not 2 or 3 such that $R(1)\neq0$. Thus, we get a partial classification of solutions to the associative Yang-Baxter equation on $M_3(F)$. For the solution, the computer algebra system Singular was involved.