Indecomposable involutive set-theoretic solutions of the Yang-Baxter equation and orthogonal dynamical extensions of cycle sets

TL;DR

This paper explores a special class of indecomposable involutive solutions to the Yang-Baxter equation using algebraic structures called left braces and cycle set extensions, focusing on solutions with specific block structures and analyzing certain left braces.

Contribution

It introduces new methods for constructing and analyzing indecomposable involutive solutions of the Yang-Baxter equation via algebraic structures like left braces and dynamical extensions.

Findings

Characterization of indecomposable involutive solutions with specific imprimitivity blocks

Analysis of one-generator left braces of multipermutation level 2

Development of new algebraic tools for Yang-Baxter solutions

Abstract

Employing the algebraic structure of the left brace and the dynamical extensions of cycle sets, we investigate a class of indecomposable involutive set-theoretic solutions of the Yang-Baxter equation having specific imprimitivity blocks. Moreover, we study one-generator left braces of multipermutation level 2.