Skew left braces of nilpotent type

Abstract

We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang-Baxter equation are explored using the structure of skew left braces.

27 pages. Accepted for publication in Proc. London Math. Soc. (3)

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• Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
• Computing skew left braces of small orders
• Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equation
• On bi-skew braces and brace blocks
• On the connection between Hopf--Galois structures and skew braces
• Set-theoretic solutions of the Yang-Baxter equation associated to weak braces
• Isoclinism of skew braces
• Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces
• Nilpotency in left semi-braces
• Inverse semi-braces and the Yang-Baxter equation
• Central nilpotency of left skew braces and solutions of the Yang-Baxter equation
• Skew braces: a brief survey
• On two-sided skew braces
• Solutions of the Yang-Baxter equation and strong semilattices of skew braces
• The structure skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation is finitely presented
• A generalization of Ito's theorem to skew braces
• Deformed solutions of the Yang-Baxter equation associated to dual weak braces
• Affine structures on groups and semi-braces
• Common divisor graphs for skew braces
• On Grün's lemma for perfect skew braces