Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs

TL;DR

This paper explores the algebraic structures arising from set-theoretic solutions to the Yang-Baxter equation, introducing a right-cyclic calculus to establish Garside structures and finite quotients analogous to Coxeter groups.

Contribution

It develops a right-cyclic calculus to analyze structure groups of involutive nondegenerate solutions, providing new proofs and insights into their Garside and I-structures.

Findings

Established Garside structures for the groups

Constructed finite quotients analogous to Coxeter groups

Provided simplified proofs for key algebraic properties

Abstract

Building on a result by W. Rump, we show how to exploit the right-cyclic law (x.y).(x.z) = (y.x).(y.z) in order to investigate the structure groups and monoids attached with (involutive nondegenerate) set-theoretic solutions of the Yang-Baxter equation. We develop a sort of right-cyclic calculus, and use it to obtain short proofs for the existence both of the Garside structure and of the I-structure of such groups. We describe finite quotients that exactly play for the considered groups the role that Coxeter groups play for Artin-Tits groups.