Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation

Abstract

We define the radical and weight of a skew left brace and provide some basic properties of these notions. In particular, we obtain a Wedderburn type decomposition for Artinian skew left braces. Furthermore, we prove analogues of a theorem of Wiegold, a theorem of Schur and its converse in the context of skew left braces. Finally, we apply these results to detect torsion in the structure group of a finite bijective non-degenerate set-theoretic solution of the Yang-Baxter equation.

15 pages

References in corpus (2)

• Semigroups of I-type
• Bi-skew braces and Hopf Galois structures

Cited by in corpus (8)

• Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation
• Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
• Nilpotency in left semi-braces
• Inverse semi-braces and the Yang-Baxter equation
• Skew braces: a brief survey
• Reflection equation as a tool for studying solutions to the Yang-Baxter equation
• The structure skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation is finitely presented
• Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation