Commutator-based Solutions to $AXA=XAX$ when $A^n=t{A}^2$

Abstract

This paper offers a novel analytical method for obtaining new non-commuting solutions for the Yang-Baxter-like matrix equation $AXA=XAX$, when the coefficient matrix $A$ satisfies the square-cyclic condition $A^n=t{A}^2$. All the consistency criteria are proved and the obtained results are demonstrated on worked examples. The findings provided by this article simultaneously generalize several existing special cases regarding this topic. Consequently, by employing the obtained results, the previously unstudied inhomogeneous Yang-Baxter matrix equation $AXA=XAX+B$ is solved when $A$ is either invertible, or satisfies $A^2=0$.