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

Patrick Dehornoy (1, 2)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 PPS - Preuves, Programmes et Systèmes

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.

Mots clés: Yang-Baxter equation; set-theoretic solution; Garside monoid; Garside group; monoid of I-type; right-cyclic law; RC-quasigroup; birack

Domaines: Théorie des groupes [math.GR]

Patrick Dehornoy. Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs. 2014. ⟨hal-00958299v3⟩