The interplay between $k$-graphs and the Yang-Baxter equation

TL;DR

This paper explores the relationship between $k$-graphs and the Yang-Baxter equation, revealing structural connections and constructing new solutions through a dual perspective approach.

Contribution

It establishes that set-theoretic solutions of the Yang-Baxter equation form a special class of single-vertex $k$-graphs and vice versa, introducing new methods to generate solutions.

Findings

Set-theoretic solutions form a special class of $k$-graphs.

Constructed an infinite family of solutions from a given one.

All single-vertex $k$-graphs correspond to YB-semigroups of involutive solutions.

Abstract

In this paper, we initiate the study of the interplay between $k$-graphs and the Yang-Baxter equation. For this, we provide two very different perspectives. One one hand, we show that the set of all set-theoretic solutions of the Yang-Baxter equation is a special class of single-vertex $k$-graphs. As a consequence, we construct an infinite family of large solutions of the Yang-Baxter equation from an arbitrarily given one. On the other hand, we prove that all single-vertex $k$-graphs are YB-semigroups of square-free, involutive solutions of the Yang-Baxter equation. Other various connections are also investigated.