Factorizations of skew braces

Abstract

We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of Itô's theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang-Baxter equation. Finally, we classify skew braces that contain no non-trivial proper ideals.

arXiv comment12 pages

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• Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation
• Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
• From Braces to Hecke algebras & Quantum Groups
• Set-theoretic solutions of the Yang-Baxter equation associated to weak braces
• Nilpotency in left semi-braces
• Inverse semi-braces and the Yang-Baxter equation
• Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices
• Schur covers of skew braces
• Near braces and p-deformed braided groups
• Novel non-involutive solutions of the Yang-Baxter equation from (skew) braces
• Solutions of the Yang-Baxter equation and strong semilattices of skew braces
• A generalization of Ito's theorem to skew braces
• Affine structures on groups and semi-braces
• Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and dynamical extensions of q-cycle sets
• Common divisor graphs for skew braces
• Algebraic structures in set-theoretic Yang-Baxter & reflection equations
• Self-distributive structures, braces & the Yang-Baxter equation