On skew braces

• Autores: Smoktunowicz, Agata; Vendramin, Claudio Leandro
• Año de publicación: 2018
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: BRACES; RADICAL RINGS; YANG-BAXTER; HOPF-GALOIS
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

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. 
Fil: Smoktunowicz, Agata. University of Edinburgh; Reino Unido
Fil: Vendramin, Claudio Leandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina