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arXiv · 2406.15800

Classification of the types for which every Hopf--Galois correspondence is bijective

Lorenzo Stefanello·Cindy Tsang
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Abstract

Let $L/K$ be any finite Galois extension with Galois group $G$. It is known by Chase and Sweedler that the Hopf--Galois correspondence is injective for every Hopf--Galois structure on $L/K$, but it need not be bijective in general. Hopf--Galois structures are known to be related to skew braces, and recently, the first-named author and Trappeniers proposed a new version of this connection with the property that the intermediate fields of $L/K$ in the image of the Hopf--Galois correspondence are in bijection with the left ideals of the associated skew brace. As an application, they classified the groups $G$ for which the Hopf--Galois correspondence is bijective for every Hopf--Galois structure on any $G$-Galois extension. In this paper, using a similar approach, we shall classify the groups $N$ for which the Hopf--Galois correspondence is bijective for every Hopf--Galois structure of type $N$ on any Galois extension.

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Lorenzo Stefanello, Cindy Tsang. 2024-06-22. Classification of the types for which every Hopf--Galois correspondence is bijective. https://doi.org/10.1016/j.jalgebra.2024.10.010

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