Extensions of set-theoretic solutions of the Yang–Baxter equation and a conjecture of Gateva-Ivanova

Vendramin, L.

doi:10.1016/j.jpaa.2015.10.018

Cited by 67 publications

(77 citation statements)

References 19 publications

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“…The seminal works of Etingof, Schedler and Soloviev [15], and Gateva-Ivanova and Van den Bergh [24], discussed algebraic and geometrical interpretations and introduced several structures associated with the class of non-degenerate involutive solutions. Such solutions have been intensively studied, see for example [17,18,19], [21,22,23], [25,26], [20], [32,34], [6], [10], [11], [13], [28], and [40].…”

Section: Introductionmentioning

confidence: 99%

Skew braces and the Yang–Baxter equation

2016

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Abstract. Braces were introduced by Rump to study non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. We generalize Rump's braces to the non-commutative setting and use this new structure to study not necessarily involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation. Based on results of Bachiller and Catino and Rizzo, we develop an algorithm to enumerate and construct classical and non-classical braces of small size up to isomorphism. This algorithm is used to produce a database of braces of small size. The paper contains several open problems, questions and conjectures.

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Section: Introductionmentioning

confidence: 99%

Skew braces and the Yang–Baxter equation

2016

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“…This gives a new example of a structural property of the solution (being MP) which can be read off its structure group (namely, its orderability). The importance of MP solutions is discussed, for instance, in [CJO10,GIC12,Ven16,BCJO17].…”

Section: Introductionmentioning

confidence: 99%

On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation

Lebed¹,

Vendramin²

2019

Proceedings of the Edinburgh Mathematical Society

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This paper explores the structure groups G (X,r) of finite nondegenerate set-theoretic solutions (X, r) to the Yang-Baxter equation. Namely, we construct a finite quotient G (X,r) of G (X,r) , generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if X injects into G (X,r) , then it also injects into G (X,r) . We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of G (X,r) . We show that multipermutation solutions are the only involutive solutions with diffuse structure group; that only free abelian structure groups are biorderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: biorderable, left-orderable, abelian, free abelian, torsion free.G (X,r) = X | xy = σ x (y)τ y (x) for all x, y ∈ X 2010 Mathematics Subject Classification. 16T25, 20N02, 06F15.

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“…Furthermore, studying this example of Vendramin one can check that it is a strong twisted union of two multipermutation solutions of multipermutation level two. Thus this yields a negative answer to a question posed by Cameron and Gateva-Ivanova [21, Open Questions 6.13 (II)(4)], although Vendramin did not notice this fact in [38].…”

Section: Introductionmentioning

confidence: 85%

A family of irretractable square-free solutions of the Yang–Baxter equation

et al. 2017

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A new family of non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation is constructed. All these solutions are strong twisted unions of multipermutation solutions of multipermutation level at most two. A large subfamily consists of irretractable and square-free solutions. This subfamily includes a recent example of Vendramin [38, Example 3.9], who first gave a counterexample to Gateva-Ivanova's Strong Conjecture [19, Strong Conjecture 2.28(I)]. All the solutions in this subfamily are new counterexamples to Gateva-Ivanova's Strong Conjecture and also they answer a question of Cameron and Gateva-Ivanova [21, Open Questions 6.13 (II)(4)]. It is proved that the natural left brace structure on the permutation group of the solutions in this family has trivial socle. Properties of the permutation group and of the structure group associated to these solutions are also investigated. In particular, it is proved that the structure groups of finite solutions in this subfamily are not poly-(infinite cyclic) groups.

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Extensions of set-theoretic solutions of the Yang–Baxter equation and a conjecture of Gateva-Ivanova

Cited by 67 publications

References 19 publications

Skew braces and the Yang–Baxter equation

Skew braces and the Yang–Baxter equation

On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation

A family of irretractable square-free solutions of the Yang–Baxter equation

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