Novel non-involutive solutions of the Yang-Baxter equation from (skew) braces
arXiv:2204.11580 · doi:10.1112/jlms.12999
Abstract
We produce novel non-involutive solutions of the Yang-Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from braces and skew braces, and surprisingly in the case of braces they are not necessarily involutive. In the case of two-sided (skew) braces one can assign such solutions to every element of the set. Novel bijective maps associated to the inverse solutions are also introduced. Moreover, we show that the recently derived Drinfeld twists of the involutive case are still admissible in the non-involutive frame and we identify the twisted -matrices and twisted coproducts. We observe, as in the involutive case that the underlying quantum algebra is not a quasi-triangular bialgebra, as one would expect, but a quasi-triangular quasi-bialgebra. The same applies to the quantum algebra of the twisted -matrices, contrary to the involutive case.
arXiv comment23 pages, LaTex. Few minor modifications in the text and some explanatory comments added
References in corpus (11)
- On set-theoretical solutions of the quantum Yang-Baxter equation
- Set-theoretical solutions to the quantum Yang-Baxter equation
- Skew braces and the Yang-Baxter equation
- On skew braces (with an appendix by N. Byott and L. Vendramin)
- On structure groups of set-theoretic solutions to the Yang-Baxter equation
- Factorizations of skew braces
- Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
- On skew braces and their ideals (with an Appendix by Agata Smoktunowicz)
- Set theoretic Yang-Baxter equation, braces and Drinfeld twists
- From Braces to Hecke algebras & Quantum Groups
- Binomial skew polynomial rings, Artin-Schelter regularity, and binomial solutions of the Yang-Baxter equation