2015/11/24 by David Bachiller, Bachiller, David, Ferran Cedó +5
Mathematics · #Rings, Modules, and Algebras #Advanced Topology and Set Theory #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1511.07769
A new family of non-degenerate involutive set-theoretic solutions of the\nYang-Baxter equation is constructed. All these solutions are strong twisted\nunions of multipermutation solutions of multipermutation level at most two. A\nlarge subfamily consists of irretractable and square-free solutions. This\nsubfamily includes a recent example of Vendramin, who first gave a\ncounterexample to Gateva-Ivanova's Strong Conjecture. All the solutions in this\nsubfamily are new counterexamples to Gateva-Ivanova's Strong Conjecture and\nalso they answer a question of Cameron and Gateva-Ivanova. It is proved that\nthe natural left brace structure on the permutation group of the solutions in\nthis family has trivial socle. Properties of the permutation group and of the\nstructure group associated to these solutions are also investigated. In\nparticular, it is proved that the structure groups of finite solutions in this\nsubfamily are not poly-(infinite cyclic) groups.\n