On Skew Braces (with an appendix by N. Byott and L. Vendramin)

Agata Smoktunowicz, L. Vendramin

Research output: Contribution to journalArticlepeer-review


Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.
Original languageEnglish
Pages (from-to)47-86
Number of pages37
JournalJournal of Combinatorial Algebra
Issue number1
Early online date8 Feb 2018
Publication statusPublished - Feb 2018


Dive into the research topics of 'On Skew Braces (with an appendix by N. Byott and L. Vendramin)'. Together they form a unique fingerprint.

Cite this