Abstract

We involve simultaneously the theory of braided groups and the theory of braces to study set-theoretic solutions of the Yang–Baxter equation (YBE). We show the intimate relation between the notions of “a symmetric group”, in the sense of Takeuchi, i.e. “a braided involutive group”, and “a left brace”. We find new results on symmetric groups of finite multipermutation level and the corresponding braces. We introduce a new invariant of a symmetric group (G,r), the derived chain of ideals ofG, which gives a precise information about the recursive process of retraction of G. We prove that every symmetric group (G,r) of finite multipermutation level m is a solvable group of solvable length ≤m. To each set-theoretic solution (X,r) of YBE we associate two invariant sequences of involutive braided groups: (i) the sequence of its derived symmetric groups; (ii) the sequence of its derived permutation groups; and explore these for explicit descriptions of the recursive process of retraction of (X,r). We find new criteria necessary and sufficient to claim that (X,r) is a multipermutation solution.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.