A family of irretractable square-free solutions of the Yang-Baxter equation. (English) Zbl 1394.16041
Summary: 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 L. Vendramin [J. Pure Appl. Algebra 220, No. 5, 2064–2076 (2016; Zbl 1337.16028), Example 3.9], who first gave a counterexample to T. Gateva-Ivanova’s strong conjecture [J. Math. Phys. 45, No. 10, 3828–3858 (2004; Zbl 1065.16037), 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 T. Gateva-Ivanova and P. Cameron [Commun. Math. Phys. 309, No. 3, 583–621 (2012; Zbl 1247.81211), 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.
MSC:
| 16T25 | Yang-Baxter equations |
| 20E22 | Extensions, wreath products, and other compositions of groups |
| 20F16 | Solvable groups, supersolvable groups |
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