Skew braces and the Yang-Baxter equation

TL;DR

This paper generalizes the concept of braces to non-commutative settings to better understand solutions to the Yang-Baxter equation, providing algorithms and a database for small structures.

Contribution

It introduces non-commutative braces, extending Rump's original concept, and develops an algorithm to enumerate and classify small braces, including non-classical ones.

Findings

Developed an enumeration algorithm for small braces

Constructed a database of braces up to isomorphism

Identified open problems and conjectures in the area

Abstract

Braces were introduced by Rump to study non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. We generalize Rump's braces to the non-commutative setting and use this new structure to study not necessarily involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation. Based on results of Bachiller and Catino and Rizzo, we develop an algorithm to enumerate and construct classical and non-classical braces of small size up to isomorphism. This algorithm is used to produce a database of braces of small size. The paper contains several open problems, questions and conjectures.