Opposite skew left braces and applications

2019/08/07 by Koch, Alan, Truman, Paul J. · 3 citations
#16T05 #16T25 #20N99 #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1908.02682

Abstract

Given a skew left brace \mathfrakB, we introduce the notion of an "opposite" skew left brace \mathfrakB', which is closely related to the concept of the opposite of a group, and provide several applications. Skew left braces are closely linked with both solutions to the Yang-Baxter Equation and Hopf-Galois structures on Galois field extensions. We show that the set-theoretic solution to the YBE given by \mathfrakB' is the inverse to the solution given by \mathfrakB; this allows us to identify the group-like elements in the Hopf algebra providing the Hopf-Galois structure using only these solutions. We also show how left ideals of \mathfrakB' correspond to the realizable intermediate fields of a certain Hopf-Galois extension of a Galois extension.

Cited by

• On cardinalities whose arithmetical properties determine the structure of solutions of the Yang--Baxter equation
• On some semidirect products of skew braces arising in Hopf-Galois theory
• On the Sylow Theorem for Skew Braces

Related

• Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures
• Almost classical skew bracoids
• Skew Braces as Remnants of Co-quasitriangular Hopf Algebras in SupLat
• Representations of skew braces
• Analogs of the lower and upper central series in skew braces: a survey