- journal article metadata
European Mathematical Society Publishing House
2018-02-11 23:30:02
Journal of Combinatorial Algebra
J. Comb. Algebra
JCA
2415-6302
2415-6310
Combinatorics
Group theory and generalizations
10.4171/JCA
http://www.ems-ph.org/doi/10.4171/JCA
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
2
2018
1
On skew braces (with an appendix by N. Byott and L. Vendramin)
Agata
Smoktunowicz
University of Edinburgh, UK
Leandro
Vendramin
University of Buenos Aires, Argentina
Braces, Yang–Baxter, rings, near-rings, triply factorized groups, matched pair of groups, bijective 1-cocycles, Hopf–Galois extensions
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.
Associative rings and algebras
Quantum theory
47
86
10.4171/JCA/2-1-3
http://www.ems-ph.org/doi/10.4171/JCA/2-1-3
2
8
2018