Retractability of set theoretic solutions of the Yang-Baxter equation

By Ferran Cedó, Eric Jespers, Jan Okninski. 2009

88 citations recorded by the source corpus.

It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presented abelian-by-finite groups (called the structure groups) are poly-${\mathbb Z}$ groups. Secondly, an example of a solution with an abelian involutive Yang-Baxter group which is not a generalized twisted union is constructed. This answers in the negative another problem of Gateva-Ivanova. The constructed solution is of multipermutation level 3. Retractability of solutions is also proved in the case where the natural generators of the IYB group are cyclic permutations. Moreover, it is shown that such solutions are generalized twisted unions.

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

View Paper: https://www.semanticscholar.org/paper/a015112b64ffcdc5fdeda5bb9a2c38a7afd93c70

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Citations in the corpus

Problems on skew left braces (2018, background): “It was proved by Cedó, Jespers and Okniński in [25] that the conjecture is true if the group G(X, r) generated by {σx : x ∈ X} is abelian.”
Extensions of set-theoretic solutions of the Yang–Baxter equation and a conjecture of Gateva-Ivanova ☆ (2015, methodology): “hematics Subject Classiﬁcation. 16T25. This work was partially supported by CONICET, PICT-2014-1376 and ICTP. 1 2 L. VENDRAMIN Multipermutation solutions has been intensively studied, see for example [4,5] and [13]. In this context, one of the most important unsolved problems is the following: Conje”
Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity (2010, background): “There are numerous works on set-theoretic solutions and related structures, of which a relevant selection for the interested reader is [39, 23, 34, 12, 27, 18, 17, 21, 24, 35, 36, 20, 21, 22, 8].”