On bi-skew braces and brace blocks

Abstract

L. N. Childs defined a bi-skew brace to be a skew brace such that if we swap the role of the two operations, then we find again a skew brace. In this paper, we give a systematic analysis of bi-skew braces. We study nilpotency and solubility, and connections between bi-skew braces and set-theoretic solutions of the Yang--Baxter equation. Further, we deal with Byott's conjecture in the case of bi-skew braces, and we use bi-skew braces as a tool to solve a classification problem proposed by L. Vendramin. In the final part, we investigate brace blocks, defined by A. Koch to be families of group operations on a given set such that any two of them yield a bi-skew brace. We provide a characterisation of brace blocks, illustrate how all known constructions in literature follow in a natural way from our characterisation, and give several new examples.

arXiv commentFinal version, published in Journal of Pure and Applied Algebra

References in corpus (14)

• On set-theoretical solutions of the quantum Yang-Baxter equation
• Set-theoretical solutions to the quantum Yang-Baxter equation
• Skew braces and the Yang-Baxter equation
• On skew braces (with an appendix by N. Byott and L. Vendramin)
• On structure groups of set-theoretic solutions to the Yang-Baxter equation
• Skew left braces of nilpotent type
• Bi-Skew Braces and Regular Subgroups of the Holomorph
• Bi-skew braces and Hopf Galois structures
• On skew braces and their ideals (with an Appendix by Agata Smoktunowicz)
• Solubility Criteria for Hopf-Galois Structures
• The Multiple Holomorphs of Finite $p$-Groups of Class Two
• The construction of multipermutation solutions of the Yang-Baxter equation of level 2
• From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, and Hopf--Galois structures
• Brace blocks from bilinear maps and liftings of endomorphisms

Cited by in corpus (5)

• On the connection between Hopf--Galois structures and skew braces
• Central nilpotency of left skew braces and solutions of the Yang-Baxter equation
• Affine structures on groups and semi-braces
• A classification of module braces over the ring of $\mathbf{p}$-adic integers
• Classification of the types for which every Hopf--Galois correspondence is bijective