# Opposite skew left braces and applications

## TL;DR

This paper introduces the concept of opposite skew left braces, explores their relationship with solutions to the Yang-Baxter Equation, and applies these ideas to Hopf-Galois structures and intermediate fields in Galois extensions.

## Contribution

It defines the opposite skew left brace and demonstrates its applications to solutions of the Yang-Baxter Equation and Hopf-Galois structures, linking algebraic and Galois-theoretic concepts.

## Key findings

- The opposite skew left brace's associated YBE solution is the inverse of the original.
- Left ideals of the opposite brace correspond to realizable intermediate fields.
- The approach simplifies identifying group-like elements in Hopf-Galois structures.

## Abstract

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.