Folding of set-theoretical solutions of the Yang-Baxter equation

TL;DR

This paper explores the relationship between invariant subsets of certain solutions to the Yang-Baxter equation and their algebraic structure, introducing the concept of foldable solutions to extend decomposability.

Contribution

It establishes a correspondence between invariant subsets and parabolic subgroups, and introduces the notion of foldable solutions in the context of the Yang-Baxter equation.

Findings

Invariant subsets correspond to parabolic subgroups with Garside structure.

Introduces foldable solutions extending decomposable solutions.

Provides a new framework for understanding solutions of the Yang-Baxter equation.

Abstract

We establish a correspondence between the invariant subsets of a non-degenerate symmetric set-theoretical solution of the quantum Yang-Baxter equation and the parabolic subgroups of its structure group, equipped with its canonical Garside structure. Moreover, we introduce the notion of a foldable solution, which extends the one of a decomposable solution.