On skew braces (with an appendix by N. Byott and L. Vendramin)

By Agata Smoktunowicz, Leandro Vendramin. 2017

193 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dbda0b863a16d19b58deeee036884d8a95a6f575

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Problems on skew left braces (2018, background): “Some work related to Problem 23 can be found in [47, 67].”
Skew left braces of nilpotent type (2018, background): “Skew braces are useful for studying regular subgroups and Hopf–Galois extensions, bijective 1-cocycles, rings and near-rings, triply factorized groups, see for example [30].”
On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation (2017, background): “Bijective group 1-cocycles have many alternative descriptions: as braided commutative groups, as skew braces, as a skew version of linear cycle sets, as certain factorized groups [18,21,27,33].”
Rota—Baxter groups, skew left braces, and the Yang—Baxter equation (2021, background): “In [40], a skew left brace (S3, ·, ◦) such that (S3, ·) is the symmetric group of order 6 and (S3, ◦) is the cyclic group of order 6 was constructed.”