Hopf Galois Structures Research Articles

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

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.

A generalization of Kummer theory to Hopf-Galois extensions

We introduce a condition for Hopf-Galois extensions that generalizes the notion of Kummer Galois extension. Namely, an H-Galois extension L/K is H-Kummer if L can be generated by adjoining to K a finite set S of eigenvectors for the action of the Hopf algebra H on L. This extends the classical Kummer condition for the classical Galois structure. With this new perspective, we shall characterize a class of H-Kummer extensions L/K as radical extensions that are linearly disjoint with the n-th cyclotomic extension of K. This result generalizes the description of Kummer Galois extensions as radical extensions of a field containing the n-th roots of the unity. The main tool is the construction of a product Hopf-Galois structure on the compositum of almost classically Galois extensions L1/K, L2/K such that L1∩M2=L2∩M1=K, where Mi is a field such that LiMi=L˜i, the normal closure of Li/K. When L/K is an extension of number or p-adic fields, we shall derive criteria on the freeness of the ring of integers OL over its associated order in an almost classically Galois structure on L/K.

Hopf-Galois structures on separable field extensions of degree pq

In 2020, Alabdali and Byott described the Hopf-Galois structures arising on Galois field extensions of squarefree degree. Extending to squarefree separable, but not necessarily normal, extensions L/K is a natural next step. One must consider now the interplay between two Galois groups G=Gal(E/K) and G′=Gal(E/L), where E is the Galois closure of L/K. In this paper, we give a characterisation and enumeration of the Hopf-Galois structures arising on separable extensions of degree pq where p and q are distinct odd primes. This work includes the results of Byott and Martin-Lyons who do likewise for the special case that p=2q+1.

Large norms in group theory

In 1935, the introduction of the norm of a group by Reinhold Baer is a turning point in group theory. In fact, Baer proved that there is a very strong relationship between the structure of the norm and that of the whole group (see [1], [2], [3], [4], [5]). Since then, the norm has been playing a very significant roles in many aspects of group theory and its applications: it has been used in [43] to describe the connection between Hopf–Galois structures and skew braces; it has been used in [23] to describe some special types of profinite groups; and it has been fundamental in the theory of subgroup lattices of groups (see [40]).In this paper, we weaken the original definition of norm by taking into account only those subgroups that are “large” in some sense. Depending on the chosen concept of largeness, the resulting norm can have an impact on the structure of the whole group that is even greater than that of Baer's norm. This is exactly what happens with the non-polycyclic norm, and in fact, Theorem 4.17 gives a precise description of generalized soluble groups in which the non-polycyclic norm is non-Dedekind (and can be considered as the main result of the paper). Other times, the resulting norms have their own peculiar behaviour; this is the case if “large” means “infinite”, “having infinite rank”, “being non-Černikov”, or “having cardinality m” for some given uncountable cardinal number m.

Skew bracoids

Skew braces are intensively studied owing to their wide ranging connections and applications. We generalize the definition of a skew brace to give a new algebraic object, which we term a skew bracoid. Our construction involves two groups interacting in a manner analogous to the compatibility condition found in the definition of a skew brace. We formulate tools for characterizing and classifying skew bracoids, and study substructures, quotients, homomorphisms, and isomorphisms. As a first application, we prove that finite skew bracoids correspond with Hopf-Galois structures on finite separable extensions of fields, generalizing the existing connection between finite skew braces and Hopf-Galois structures on finite Galois extensions.

Non‐abelian simple groups which occur as the type of a Hopf–Galois structure on a solvable extension

AbstractWe determine the finite non‐abelian simple groups which occur as the type of a Hopf–Galois structure on a solvable extension. In the language of skew braces, our result gives a complete list of finite non‐abelian simple groups which occur as the additive group of a skew brace with solvable multiplicative group.

Nonabelian Case of Hopf Galois Structures on Nonnormal Extensions of Degree pqw

We look at Hopf Galois structures with square free pqw degree on separable field extensions (nonnormal) L/K. Where E/K is the normal closure of L/K, the group permutation of degree pqw is G = Gal(E/K). We study details of the nonabelian case, where Jl = ⟨σ, [τ, αl ]⟩ is a nonabelian regular subgroup of Hol(N) for 1 ≤ l ≤ w − 1. We first find the group permutation G, and then the Hopf Galois structures for each G. In this case, there exists four G such that the Hopf Galois structures are admissible within the field extensions L/K.

Inducing braces and Hopf Galois structures

Let p be a prime number and let n be an integer not divisible by p and such that every group of order np has a normal subgroup of order p. (This holds in particular for p>n.) Under these hypotheses, we obtain a one-to-one correspondence between the isomorphism classes of braces of size np and the set of pairs (Bn,[τ]), where Bn runs over the isomorphism classes of braces of size n and [τ] runs over the classes of group morphisms from the multiplicative group of Bn to Zp⁎ under a certain equivalence relation. This correspondence gives the classification of braces of size np from the one of braces of size n. From this result we derive a formula giving the number of Hopf Galois structures of abelian type Zp×E on a Galois extension of degree np in terms of the number of Hopf Galois structures of abelian type E on a Galois extension of degree n. For a prime number p≥7, we apply the obtained results to describe all left braces of size 12p and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree 12p.

On the connection between Hopf–Galois structures and skew braces

AbstractWe present a different version of the well‐known connection between Hopf–Galois structures and skew braces, building on a recent paper of A. Koch and P. J. Truman. We show that the known results that involve this connection easily carry over to this new perspective, and that new ones naturally appear. As an application, we present new insights on the study of the surjectivity of the Hopf–Galois correspondence, explaining in more detail the role of bi‐skew braces in Hopf–Galois theory.

On ρ-conjugate Hopf–Galois structures

AbstractThe Hopf–Galois structures admitted by a Galois extension of fields $ L/K $ with Galois group G correspond bijectively with certain subgroups of $ \mathrm{Perm}(G) $. We use a natural partition of the set of such subgroups to obtain a method for partitioning the set of corresponding Hopf–Galois structures, which we term ρ-conjugation. We study properties of this construction, with particular emphasis on the Hopf–Galois analogue of the Galois correspondence, the connection with skew left braces, and applications to questions of integral module structure in extensions of local or global fields. In particular, we show that the number of distinct ρ-conjugates of a given Hopf–Galois structure is determined by the corresponding skew left brace, and that if $ H, H^{\prime} $ are Hopf algebras giving ρ-conjugate Hopf–Galois structures on a Galois extension of local or global fields $ L/K $ then an ambiguous ideal $ \mathfrak{B} $ of L is free over its associated order in H if and only if it is free over its associated order in Hʹ. We exhibit a variety of examples arising from interactions with existing constructions in the literature.

The ring of integers of Hopf-Galois degree p extensions of p-adic fields with dihedral normal closure

For an odd prime number p, we consider degree p extensions L/K of p-adic fields with normal closure L˜ such that the Galois group of L˜/K is the dihedral group of order 2p. We shall prove a complete characterization of the freeness of the ring of integers OL over its associated order AL/K in the unique Hopf-Galois structure on L/K, which is analogous to the one already known for cyclic degree p extensions of p-adic fields. We shall derive positive and negative results on criteria for the freeness of OL as AL/K-module.

Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group

Let $G$ and $N$ be two finite groups of the same order. It is well-known that the existences of the following are equivalent: (a) a Hopf-Galois structure of type $N$ on any Galois $G$-extension; (b) a skew brace with additive group $N$ and multiplicative group $G$; (c) a regular subgroup isomorphic to $G$ in the holomorph of $N$. We shall say that $(G,N)$ is realizable when any of the above exists. Fixing $N$ to be a cyclic group, W. Rump (2019) has determined the groups $G$ for which $(G,N)$ is realizable. In this paper, fixing $G$ to be a cyclic group instead, we shall give a complete characterization of the groups $N$ for which $(G,N)$ is realizable.

Hopf-Galois module structure of quartic Galois extensions of Q

Given a quartic Galois extension $L/\mathbb{Q}$ of number fields and a Hopf-Galois structure $H$ on $L/\mathbb{Q}$, we study the freeness of the ring of integers $\mathcal{O}_L$ as module over the associated order $\mathfrak{A}_H$ in $H$. For the classical Galois structure $H_c$, we know by Leopoldt's theorem that $\mathcal{O}_L$ is $\mathfrak{A}_{H_c}$-free. If $L/\mathbb{Q}$ is cyclic, it admits a unique non-classical Hopf-Galois structure, whereas if it is biquadratic, it admits three such Hopf-Galois structures. In both cases, we obtain that freeness depends on the solvability in $\mathbb{Z}$ of certain generalized Pell equations. We shall translate some results on Pell equations into results on the $\mathfrak{A}_H$-freeness of $\mathcal{O}_L$.

Skew left braces and isomorphism problems for Hopf–Galois structures on Galois extensions

Given a finite group [Formula: see text], we study certain regular subgroups of the group of permutations of [Formula: see text], which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to [Formula: see text] and Hopf–Galois structures admitted by a Galois extension of fields with Galois group isomorphic to [Formula: see text]. We study the questions of when two such subgroups yield isomorphic skew left braces or Hopf–Galois structures involving isomorphic Hopf algebras. In particular, we show that in some cases the isomorphism class of the Hopf algebra giving a Hopf–Galois structure is determined by the corresponding skew left brace. We investigate these questions in the context of a variety of existing constructions in the literature. As an application of our results we classify the isomorphically distinct Hopf algebras that give Hopf–Galois structures on a Galois extension of degree [Formula: see text] for [Formula: see text] prime numbers.

Hopf-Galois structures on non-normal extensions of degree related to Sophie Germain primes

We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions L / K of squarefree degree n . If E / K is the normal closure of L / K then G = Gal ( E / K ) can be viewed as a permutation group of degree n . We show that G has derived length at most 4, but that many permutation groups of squarefree degree and of derived length 2 cannot occur. We then investigate in detail the case where n = p q where q ≥ 3 and p = 2 q + 1 are both prime. (Thus q is a Sophie Germain prime and p is a safeprime). We list the permutation groups G which can arise, and we enumerate the Hopf-Galois structures for each G . There are six such G for which the corresponding field extensions L / K admit Hopf-Galois structures of both possible types.

Hopf-Galois algebras and their Poisson structures

Hopf-Galois objects are natural generalizations of Hopf algebras with a Galois-theoretic flavor. In this paper, we prove a criterion for an Ore extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, introduce a concept of Poisson Hopf-Galois algebra and establish a relationship between Poisson Hopf-Galois algebras and Poisson Hopf algebras. More importantly, we study Poisson Hopf-Galois structures on Poisson polynomial algebras, and give a necessary and sufficient condition for the Poisson enveloping algebra of a Poisson Hopf-Galois algebra to be a Hopf-Galois algebra.

Induced Hopf Galois structures and their local Hopf Galois modules

The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as the direct product of the corresponding regular groups of the inducing subextensions. We describe here the associated Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. In order to deal with their associated orders we develop a general method to compute bases and free generators in terms of matrices coming from representation theory of Hopf modules. In the case of an induced Hopf Galois structure it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.

On [formula omitted]-Hopf-Galois structures

Let K/F be a finite Galois extension of fields with Gal(K/F)=Γ. In an earlier work of Timothy Kohl, the author enumerated dihedral Hopf-Galois structures acting on dihedral extensions. The dihedral group is one particular example of a semidirect product of Zn and Z2. In this article we count the number of Hopf-Galois structures with Galois group Γ of type G, where Γ,G are groups of the form Zn⋊ϕZ2 when n is odd with radical of n being a Burnside number. As an application we also find the corresponding number of skew braces.

Hopf–Galois structures on ambiskew polynomial rings

We provide necessary and sufficient conditions to extend the Hopf–Galois algebra structure on an algebra R to a generalized ambiskew ring based on R , in a way such that the added variables for the extension are skew-primitive in an appropriate sense. We show that the associated Hopf algebra is again a generalized ambiskew ring, based on a suitable Hopf algebra \underline{H}(R) . Several examples are examined, including the Hopf–Galois objects over U_q(\mathfrak{sl}_2) .

From endomorphisms to bi-skew braces, regular subgroups, the Yang–Baxter equation, and Hopf–Galois structures

The interplay between set-theoretic solutions of the Yang–Baxter equation of Mathematical Physics, skew braces, regular subgroups, and Hopf–Galois structures has spawned a considerable body of literature in recent years.In a recent paper, Alan Koch generalised a construction of Lindsay N. Childs, showing how one can obtain bi-skew braces (G,⋅,∘) from an endomorphism of a group (G,⋅) whose image is abelian.In this paper, we characterise the endomorphisms of a group (G,⋅) for which Koch's construction, and a variation on it, yield (bi-)skew braces. We show how the set-theoretic solutions of the Yang–Baxter equation derived by Koch's construction carry over to our more general situation, and discuss the related Hopf–Galois structures.