Solutions to a system of Yang-Baxter matrix equations

TL;DR

This paper investigates solutions to a generalized Yang-Baxter matrix system, exploring geometric, topological, and algebraic aspects, including special cases like idempotent orthogonal complements and solutions for 2x2 matrices.

Contribution

It introduces a comprehensive analysis of a generalized Yang-Baxter matrix system, including existence conditions, special case characterizations, and solutions for small matrices.

Findings

Existence conditions for doubly stochastic solutions

Characterization of solutions when A and B are idempotent orthogonal complements

Complete solution set for 2x2 matrices using algebraic techniques

Abstract

In this article, a system of Yang-Baxter-type matrix equations is studied, $XAX=BXB$, $XBX=AXA$, which "generalizes" the matrix Yang-Baxter equation and exhibits a broken symmetry. We investigate the solutions of this system from various geometric and topological points of view. We analyze the existence of doubly stochastic solutions and intertwining solutions to the system and describe the conditions for their existence. Furthermore, we characterize the case when $A$ and $B$ are idempotent orthogonal complements. i.e., $A^2 =A, B^2= B, AB = BA =0$. We also completely characterize the set of solutions for $n=2$ using commutative algebraic techniques.