Skew braces and the Galois correspondence for Hopf Galois structures

Abstract

Let L/K be a Galois extension of fields with Galois group Γ, and suppose L/K is also an H-Hopf Galois extension. Using the recently uncovered connection between Hopf Galois structures and skew left braces, we introduce a method to quantify the failure of surjectivity of the Galois correspondence from subHopf algebras of H to intermediate subfields of L/K, given by the Fundamental Theorem of Hopf Galois Theory. Suppose L ⊗ K H = LN where N ∼ = (G, ⋆). Then there exists a skew left brace (G, ⋆, •) where (G, •) ∼ = Γ. We show that there is a bijective correspondence between the set of intermediate fields E between K and L that correspond to K-subHopf algebras of H and a set of subskew left braces of G that we call the •-stable subgroups of (G, ⋆). Counting these subgroups and comparing that number with the number of subgroups of Γ ∼ = (G, •) describes how far the Galois correspondence for the H-Hopf Galois structure is from being surjective. The method is illustrated by a variety of examples.

On skew braces and their ideals

“…Skew braces have connections to several different topics, see for example [4,12,14,20,21,27,33]. In particular, skew braces provide the right algebraic framework to study set-theoretic solutions to the Yang-Baxter equation.…”

Bounds on the number of ideals in finite commutative nilpotent $\mathbb{F}_p$-algebras

“…As observed in [Ch17], that comparison implies immediately that if A 2 = 0, then there are subspaces of A that are not ideals, and hence the Galois correspondence cannot be surjective.…”

On the Galois correspondence for Hopf Galois structures arising from finite radical algebras and Zappa-Szép products

“…But except for the Greither-Pareigis examples, very little was known about the image of the Galois correspondence for an H-Hopf Galois structure on a G-Galois extension L/K of fields until it was observed in [11] that the K-subHopf algebras of H correspond bijectively to the subgroups N of N that are normalized by λ(G). Using that result, the paper [6] examined the G-Galois extensions L/K, where G is an abelian p-group of order p n and L/K has an H-Hopf Galois structure of type N , also an abelian p-group of order p n . If H has type N , then there is a regular embedding of G into Hol(N ): call the image T .…”