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

By Tatiana Gateva-Ivanova, Shahn Majid. 2005

104 citations recorded by the source corpus.

We 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 characterisation 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 rS on S(X,r) as a set. 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⋈S⋈S equivalent to rS a solution, and extensions (S⋈T,rS⋈T). Examples include a general ‘double’ construction (S⋈S,rS⋈S) and some concrete extensions, their actions and graphs based on small sets.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/82bd3c92fbf780d3b9b49b48bcc9745895f531a5

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Citations in the corpus

Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity (2010, methodology): “EVA-IVANOVA In section 5 we prove Theorem 1.2. The proof involves the results of sections 3, 4, and results on binomial skew polynomial rings and set-theoretic solutions of YBE from [25], [18], [17], [21]. 2. Preliminaries - some definitions and facts In this section we recall basic notions and resu”
Solutions of the Yang-Baxter equation associated with a left brace (2015, background): “A subclass of this type of solutions, the non-degenerate involutive ones, has received a lot of attention in the last years [6, 7, 8, 11, 13, 14, 15, 16, 18, 19, 21, 22, 23].”
Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation (2016, background): “This class of solutions has received a lot of attention in recent years, see for example [8, 9, 12, 13, 14, 17, 18, 19, 20, 21, 25, 26, 29, 31, 39].”
A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation (2004, methodology): “We also state a recent result from [15], in which matched pairs approach is used to describe extensions of solutions.”
Involutive Yang-Baxter groups (2008, methodology): “of the braided equation (or a solution of the Yang-Baxter equation, in the terminology used for example in [6, 7]) r12r23r12 = r23r12r23.”
On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation (2017, background): “Bijective group 1-cocycles have many alternative descriptions: as braided commutative groups, as skew braces, as a skew version of linear cycle sets, as certain factorized groups [18,21,27,33].”
Retractability of set theoretic solutions of the Yang-Baxter equation (2009, unknown): “G R ] 2 3”
Braces and the Yang–Baxter Equation (2012, background): “Recently many results on set-theoretic solutions appeared in several papers by Etingof, Schedler and Soloviev [11], Gateva-Ivanova [13], Gateva-Ivanova and Van den Bergh [16], Lu, Yan and Zhu [24], Rump [31, 33, 36], Jespers and Okniński [20, 21], Cedó, Jespers and Okniński [7], Cedó, Jespers and de”
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].”
Multipermutation Solutions of the Yang–Baxter Equation (2009, methodology): “rresponding factorisation of S(X,r), G(X,r), and G(X,r), see Theorems 4.16, 4.19 and 4.21. We make essential use of the matched pairs approach to solutions (in the most general setting), developed in [GIM08]. Section 5 continues the study of multipermutation solutions of low levels which was initiat”