On skew braces (with an appendix by N. Byott and L. Vendramin)
type: Item
instance of: scholarly article
title: On skew braces (with an appendix by N. Byott and L. Vendramin)
zbMATH Open document ID: 1416.16037
DOI: 10.4171/JCA/2-1-3
author: Agata Smoktunowicz
author: L. Vendramin
published in: Journal of Combinatorial Algebra
publication date: 2018-04-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1705.06958
review text: Summary: Braces are generalizations of radical rings, introduced by \textit{W. Rump} [J. Algebra 307, No. 1, 153--170 (2007; Zbl 1115.16022)] 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.
Mathematics Subject Classification ID: 16T25
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 6857320
zbMATH Keywords: braces
zbMATH Keywords: Yang-Baxter
zbMATH Keywords: rings
zbMATH Keywords: near-rings
zbMATH Keywords: triply factorized groups
zbMATH Keywords: matched pair of groups
zbMATH Keywords: bijective 1-cocycles
zbMATH Keywords: Hopf-Galois extensions
describes a project that uses: Magma
describes a project that uses: YangBaxter
MaRDI profile type: Publication
arXiv ID: 1705.06958
cites work: Q3134546
cites work: Extensions, matched products, and simple braces
cites work: Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks
cites work: Counterexample to a conjecture about braces.
cites work: Finite group factorizations and braiding
cites work: The Magma algebra system. I: The user language
cites work: Uniqueness of hopf galois structure for separable field extensions
cites work: HOPF–GALOIS STRUCTURES ON FIELD EXTENSIONS WITH SIMPLE GALOIS GROUPS
cites work: Hopf-Galois structures on Galois field extensions of degree \(pq\).
cites work: Solubility criteria for Hopf-Galois structures
cites work: On the Yang-Baxter equation and left nilpotent left braces
cites work: Involutive Yang-Baxter groups
cites work: Retractability of set theoretic solutions of the Yang-Baxter equation
cites work: Braces and the Yang-Baxter equation
cites work: On the hopf galois theory for separable field extensions
cites work: Q4495793
cites work: Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs.
cites work: On some unsolved problems in quantum group theory
cites work: A method of construction of finite-dimensional triangular semisimple Hopf algebras
cites work: Biperfect Hopf algebras.
cites work: Set-theoretical solutions to the quantum Yang-Baxter equation
cites work: Indecomposable set-theoretical solutions to the quantum Yang-Baxter equation on a set with a prime number of elements
cites work: BIQUANDLES AND THEIR APPLICATION TO VIRTUAL KNOTS AND LINKS
cites work: A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation
cites work: Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity
cites work: Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups
cites work: Multipermutation solutions of the Yang-Baxter equation
cites work: Set-theoretic solutions of the Yang-Baxter equation, graphs and computations
cites work: Matched pairs approach to set theoretic solutions of the Yang-Baxter equation.
cites work: Semigroups of \(I\)-type
cites work: Hopf Galois theory for separable field extensions
cites work: Skew braces and the Yang–Baxter equation
cites work: Q3440009
cites work: Q4317887
cites work: Classification of the Hopf Galois structures on prime power radical extensions
cites work: On the set-theoretical Yang-Baxter equation
cites work: Q3807388
cites work: Q3111409
cites work: Nilpotent Numbers
cites work: A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation
cites work: Braces, radical rings, and the quatum Yang-Baxter equation.
cites work: Classification of cyclic braces.
cites work: SEMIDIRECT PRODUCTS IN ALGEBRAIC LOGIC AND SOLUTIONS OF THE QUANTUM YANG–BAXTER EQUATION
cites work: Q5253614
cites work: On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation
cites work: Non-unitary set-theoretical solutions to the quantum Yang-Baxter equation.
cites work: Products of groups and local nearrings
cites work: Matched pairs of groups and bismash products of hopf algebras
cites work: Q4650810
cites work: Classical solutions of the quantum Yang-Baxter equation
OpenAlex ID: W3101270604
description: scientific article