Set theoretic Yang-Baxter equation, braces and Drinfeld twists

TL;DR

This paper explores the algebraic structures called braces that generate set-theoretic solutions to the Yang-Baxter equation, and extends the understanding of their relation to Drinfeld twists and Baxterized solutions.

Contribution

It provides a comprehensive expression of set-theoretic solutions in terms of admissible Drinfeld twists, extending previous preliminary results.

Findings

Identifies the generic form of twists for set-theoretic solutions

Shows these twists are admissible satisfying a co-cycle condition

Extends results to Baxterized solutions derived from set-theoretic solutions

Abstract

We consider involutive, non-degenerate, finite set theoretic solutions of the Yang-Baxter equation. Such solutions can be always obtained using certain algebraic structures that generalize nil potent rings called braces. Our main aim here is to express such solutions in terms of admissible Drinfeld twists substantially extending recent preliminary results. We first identify the generic form of the twists associated to set theoretic solutions and we show that these twists are admissible, i.e. they satisfy a certain co-cycle condition. These findings are also valid for Baxterized solutions of the Yang-Baxter equation constructed from the set theoretical ones.