What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? Is it possible to write set-theoretical solutions of Quantum Yang-Baxter equations as matrices? Thank you very much.



What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE?
Let (X,S) be a set-theoretical solution of the QYBE, where X is a set and S a bijection of XxX. If (X,S) satisfies some additional properties (non-degenerate, involutive and braided), then it defines an invertible operator R that satisfies the QYBE. 
Not all the R matrices that are solutions of the QYBE are obtained in this way.


I advice you to read the paper of P.Etingof, T.Schedler and A.Soloviev on set-theoretical solutions and the book of E.Jespers and J.Okninski 'Noetherian semigroup algebras' section 8. see also T. Gateva-Ivanova papers on the topic.


