Multipermutation Solutions of the Yang–Baxter Equation

By Tatiana Gateva-Ivanova, Peter J. Cameron. 2009

102 citations recorded by the source corpus.

Set-theoretic solutions of the Yang–Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set X and a function r : X × X → X × X which satisfies the braid relation.We examine solutions here mainly from the point of view of permutation groups: a solution gives rise to a map from X to the symmetric group Sym(X) on X satisfying certain conditions, whose image we call a Yang–Baxter permutation group. Our results include new constructions based on strong twisted unions, with an investigation of retracts and the multipermutation level and the solvable length of the groups defined by the solutions; new results about decompositions of solutions of arbitrary cardinality into invariant subsets and decompositions and factorisations of the associated Yang–Baxter group as a product of groups of the solutions defined by these invariant subsets. In particular, we obtain strong decomposition results if the Yang–Baxter permutation group is abelian or the solution is of finite multipermutation level.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ad4393a26bdfb92001120eda06de30ba108f43e1

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Citations in the corpus

Skew left braces of nilpotent type (2018, result): “Despite the conjecture was proved to be false [32], it was the motivation of several original investigations [8, 9, 14, 26].”
On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation (2017, background): “The importance of MP solutions is discussed, for instance, in [2,7,17,36].”
Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity (2010, background): “As in [18, 20, 21, 22, 23], to each quadratic set (X, r) we associate canonically algebraic objects (see Definition 2.”
Problems on skew left braces (2018, unknown): “Later in [41] and in [26] it was shown using other methods that the result is also valid if G(X, r) is infinite abelian.”
Braces and the Yang–Baxter Equation (2012, methodology): “The subgroup of SymX generated by {σx | x ∈ X} is denoted by G(X, r) (see [14]).”
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation (2015, methodology): “We disprove the relation Nm = 2 m−1+1 conjectured by Cameron and Gateva-Ivanova [22], using a computer-aided computation of the Nm for small m. Finally, we express the second cohomology of LND braided sets—in particular, cycle sets—in terms of the first group cohomology of their structure groups (Th”
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].”
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].”
Extensions of set-theoretic solutions of the Yang–Baxter equation and a conjecture of Gateva-Ivanova ☆ (2015, background): “Multipermutation solutions has been intensively studied, see for example [4,5] and [13].”
On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation (2015, background): “There are many interesting results in this area [10, 11, 12, 13, 18, 20, 19, 22, 27, 44, 29, 55].”