2015/07/09 by Tatiana Gateva-Ivanova, Gateva-Ivanova, Tatiana · 6 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1507.02602
We involve simultaneously the theory of matched pairs of groups and the\ntheory of braces to study set-theoretic solutions of the Yang-Baxter equation\n(YBE). We show the intimate relation between the notions of a symmetric group\n(a braided involutive group) and a left brace, and find new results on\nsymmetric groups of finite multipermutation level and the corresponding braces.\nWe introduce a new invariant of a symmetric group (G,r), \the derived\nchain of ideals of G, which gives a precise information about the recursive\nprocess of retraction of G. We prove that every symmetric group (G,r) of\nfinite multipermutation level m is a solvable group of solvable length at\nmost m. To each set-theoretic solution (X,r) of YBE we associate two\ninvariant sequences of symmetric groups: (i) the sequence of its derived\nsymmetric groups; (ii) the sequence of its derived permutation groups and\nexplore these for explicit descriptions of the recursive process of retraction.\nWe find new criteria necessary and sufficient to claim that (X, r) is a\nmultipermutation solution.\n