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

By Leandro Vendramin. 2015

71 citations recorded by the source corpus.

Abstract We develop a theory of extensions for involutive and nondegenerate solutions of the set-theoretic Yang–Baxter equation and use it to produce new families of solutions. As an application we construct an infinite family of counterexamples to a conjecture of Gateva-Ivanova related to the retractability of square-free solutions.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/676331c8a4eb5383ceb9daae1509636e28a8192c

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation (2016, background): “s involutive if r2 = id X2, and it is non-degenerate if fx and gx are bijective maps from Xto X, for all x∈ X. 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]. Braces were introduced to study this typ”
Problems on skew left braces (2018, methodology): “A theory of dynamical extensions of cycle sets was used in [69] to produce a counterexample to Gateva–Ivanova conjecture.”
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation (2015, methodology): “is a construction of new examples of racks and braided sets via an extension procedure using cocycles of low degree. Recently, the extension techniques were adapted to cycle sets by the second author [37], resulting in counter-examples to several conjectures concerning involutive solutions to the Ya”
On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation (2015, methodology): “tion is also called a multipermutation solution, that is a solution which has a ﬁnite multipermutation level (for a detailed deﬁnition see [23], [18]). There are many interesting results in this area [10, 11, 12, 13, 18, 20, 19, 22, 27, 44, 29, 55]. Proposition5.16from [23], Proposition 7 [44] and t”
Skew left braces of nilpotent type (2018, background): “ateva–Ivanova’s strong conjecture [13, 2.28(I)] states that square-free involutive non-degenerate set-theoretic solutions are multipermutation solutions. Despite the conjecture was proved to be false [32], it was the motivation of several original investigations [8, 9, 14, 26]. Braces provide a powe”