Classification of constant solutions for associative Yang-Baxter equation on $gl(3)$

TL;DR

This paper classifies all non-equivalent constant solutions to the associative Yang-Baxter equation on gl(3), revealing new algebraic structures such as quadratic trace Poisson brackets and anti-Frobenius algebras.

Contribution

It provides a complete classification of solutions for gl(3), introducing new algebraic structures and examples not previously documented.

Findings

All non-equivalent solutions for gl(3) identified

New quadratic trace Poisson brackets constructed

Discovery of anti-Frobenius associative algebras

Abstract

We find all non-equivalent constant solutions for classical associative Yang-Baxter equation for $gl(3)$. New examples found in the classification yield the corresponding quadratic trace Poisson brackets, double Poisson structures on free associative algebra with three generators and anti-Frobenius associative algebras.