Radical and weight of skew braces and their applications to structure groups of solutions of the Yang–Baxter equation
Advances in Mathematics(2021)
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.
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