A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation

By Tatiana Gateva-Ivanova. 2004

137 citations recorded by the source corpus.

A bijective map r: X2→X2, where X={x1,…,xn} is a finite set, is called a set-theoretic solution of the Yang–Baxter equation (YBE) if the braid relation r12r23r12=r23r12r23 holds in X3. A nondegenerate involutive solution (X,r) satisfying r(xx)=xx, for all x∈X, is called square-free solution. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions (X,r) and the associated Yang–Baxter algebraic structures—the semigroup S(X,r), the group G(X,r) and the k-algebra A(k,X,r) over a field k, generated by X and with quadratic defining relations naturally arising and uniquely determined by r. We study the properties of the associated Yang–Baxter structures, and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, ...

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a0323d01ad94f8b6f7ed74e25d3049efff7fa51f

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Citations in the corpus

Matched pairs approach to set theoretic solutions of the Yang–Baxter equation (2005, methodology): “As in [8] to each quadratic map r : X2 −→ X2 we associate canonically algebraic objects (see Definition 2.”
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].”
Garside Groups and Yang–Baxter Equation (2009, background): “Indeed, she shows in [ 13 ] that the monoid corresponding to a special case of non-degenerate, involutive, and braided set-theoretical solution (square-free) has a structure of distributive lattice with respect to left and right divisibility and that the left least common multiple of the generators ”
Monoids and Groups of I-Type (2005, unknown): “It then is shown by Gateva-Ivanova [ 10 ] that every (square-free) nondegenerate monoid of skew type that is a solution of the quantum Yang–Baxter equation is binomial.”
Extensions of set-theoretic solutions of the Yang–Baxter equation and a conjecture of Gateva-Ivanova ☆ (2015, background): “The theory of involutive solutions was further developed in several papers such as [11, 12], [18,19], [13], [3], [14,15], [4,5], [8] and [7].”
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].”
Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity (2010, methodology): “A class of PBW Arin Schelter regular rings of arbitrarily high global dimension n, were introduced and studied in [15], [25], [18], [17].”
Retractability of set theoretic solutions of the Yang-Baxter equation (2009, unknown): “G R ] 2 3”
Involutive Yang-Baxter groups (2008, background): “Keywords: Yang-Baxter equation, involutive non-degenerate solutions, group of I-type, finite solvable group. received a lot of attention by Etingof, Schedler and Soloviev [5], Gateva-Ivanova and Van den Bergh [6, 7], Rump [13], Jespers and Okninski [8, 9] and others.”
Braces and the Yang–Baxter Equation (2012, methodology): “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 an”
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].”