Matched pairs approach to set-theoretic solutions of the Yang-Baxter equation

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dc.creatorGateva-Ivanova, Tatiana
dc.creatorMajid, Shahn
dc.date2005-07-19
dc.date2007-01-30
dc.date.accessioned2026-07-07T07:43:33Z
dc.date.available2026-07-07T07:43:33Z
dc.descriptionWe study set-theoretic solutions $(X,r)$ of the Yang-Baxter equations on a set $X$ in terms of the induced left and right actions of $X$ on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid $S(X,r)$ and we show that $r$ extends as a solution $(S(X,r),r_S)$. Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids $S$ of the required type, including iterated products $S\bowtie S\bowtie S$ equivalent to $r_S$ a solution, and extensions $(S\bowtie T,r_{S\bowtie T})$. Examples include a general `double' construction $(S\bowtie S,r_{S\bowtie S})$ and some concrete extensions, their actions and graphs based on small sets.
dc.description58 pages 4 figures. Overhauled with new cleaner results and much extended section 4
dc.identifierhttps://arxiv.org/abs/math/0507394
dc.identifierhttp://arxiv.org/abs/math/0507394
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/122869
dc.subjectQuantum Algebra
dc.subjectCombinatorics
dc.subjectRings and Algebras
dc.titleMatched pairs approach to set-theoretic solutions of the Yang-Baxter equation
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