Notions of equivalence among constant Yang--Baxter Operators: To $\infty$-Equivalence and Beyond

Abstract

Recent advances in the classification of certain classes of Yang-Baxter operators through a holistic categorical approach has some of us wondering about ways to identify solutions. In particular the classification of charge conserving solutions in all dimensions inspires the question: \emph{what fraction of solutions are \textbf{close to} charge conserving solutions, under any \textbf{reasonable} notion of equivalence?} Of course \emph{reasonable} and \emph{close to} are open to interpretation. Here we suggest some precise notions of both, and prove that in dimension $2$, 10 out of 11 local families of solutions are close to charge conserving, and the one that misses is a singleton.

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