Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova

TL;DR

This paper develops a theory for extending solutions to the set-theoretic Yang-Baxter equation, producing new solutions and counterexamples to a conjecture about their properties, advancing understanding in algebraic structures.

Contribution

It introduces a novel extension framework for involutive and nondegenerate solutions, leading to new solutions and counterexamples to an existing conjecture.

Findings

Constructed an infinite family of counterexamples to Gateva-Ivanova's conjecture

Developed a systematic method for extending solutions of the Yang-Baxter equation

Produced new classes of solutions with specific algebraic properties

Abstract

We develop a theory of extensions for involutive and nondegenerate solutions of the set-theoretic Yang-Baxter equation and use it to produce new families of solutions. As an application we construct an infinite family of counterexamples to a conjecture of Gateva-Ivanova related to the retractability of square-free solutions.