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.
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asked Feb 24, 2014 at 14:07
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$\begingroup$ See mathoverflow.net/questions/201876/yang-baxter-explanation/… to see some information related to set-theoretical solutions. $\endgroup$– Leandro VendraminCommented Feb 22, 2016 at 21:04
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1 Answer
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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.
