Opposite skew left braces and applications

By Alan Koch, Paul J. Truman. Journal of Algebra, 2019

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Given a skew left brace $\mathfrak{B}$, we introduce the notion of an "opposite" skew left brace $\mathfrak{B}'$, 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 $\mathfrak{B}'$ is the inverse to the solution given by $\mathfrak{B}$; 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 $\mathfrak{B}'$ correspond to the realizable intermediate fields of a certain Hopf-Galois extension of a Galois extension.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e0d1f3a17f25bf9a25c2e18de0e0d84794001937