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

By Patrick Dehornoy. 2014

96 citations recorded by the source corpus.

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 play for the considered groups the role that Coxeter groups play for Artin–Tits groups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4aae4025c8c85aa40a5f4f77ea99bf51e498876f

How researchers see this paper

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In Geometry and Topology

1. Leandro Vendramin
2. Agata Smoktunowicz
3. Victoria Lebed
4. Ferran Cedó

In Mathematical Physics

1. Victoria Lebed
2. Leandro Vendramin
3. Agata Smoktunowicz

Citations in the corpus

• On skew braces (with an appendix by N. Byott and L. Vendramin) (2017, background): “Involutive solutions have been intensively studied; see for example [14, 18, 25, 26, 28, 29, 30].”
• On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation (2017, result): “The quotient G(X,r) is in this case very close to that from [Deh15].”