In Hopf-Galois theory, every $H$-Hopf-Galois structure on a field extension $K/k$ gives rise to an injective map $\mathcal{F}$ from the set of $k$-sub-Hopf algebras of $H$ into the intermediate fields of $K/k$. Recent papers on the failure of the surjectivity of $\mathcal{F}$ reveal that there exist many Hopf-Galois structures for which there are many more subfields than sub-Hopf algebras. This paper surveys and illustrates group-theoretical methods to determine $H$-Hopf-Galois structures on finite separable extensions in the extreme situation when $H$ has only two sub-Hopf algebras.
The 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 f…
We investigate the theory of Hopf-Galois extensions for monoidal Hom-Hopf algebras. As the main result of this paper, we prove the Schneider's affineness theorems in the case of monoidal Hom-Hopf algebras in terms of the theory of the total integral and Hom-Hopf Galois extensions. In addition, we obtain the affineness criterion for relative Hom-Hopf module associated with faithfully flat Hom-Hopf Galois extensions.
By using the language of cogroupoids, we show that Hopf-Galois objects of a twisted Calabi-Yau Hopf algebra with bijective antipode are still twisted Calabi-Yau, and give their Nakayama automorphism explicitly. As applications, cleft Galois objects of twisted Calabi-Yau Hopf algebras and Hopf-Galois objects of the quantum automorphism groups of non-degenerate bilinear forms are proved to be twisted Calabi-Yau.
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 $G=\mathrm{Gal}(E/K)$ 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 $n=pq$ where $q \geq 3$ and $p=2q+1$ are both prime. (Thus $q$ is a Sophie Germain prime and $p$ is a safeprime). We list the permutation groups $G$ which can arise, a…
We develop Hopf-Galois theory for weak Hopf algebras, and recover analogs of classical results for Hopf algebras. Our methods are based on the recently introduced Galois theory for corings. We focus on the situatation where the weak Hopf algebra is a groupoid algebra or its dual. We obtain weak versions of strongly graded rings, and classical Galois extensions.
We 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.
Let $L/F$ be a Galois extension of fields with Galois group isomorphic to the quaternion group of order $ 8 $. We describe all of the Hopf-Galois structures admitted by $ L/F $, and determine which of the Hopf algebras that appear are isomorphic as Hopf algebras. In the case that $ F $ has characteristic zero we also determine which of these Hopf algebras are isomorphic as $ F $-algebras and explicitly compute their Wedderburn-Artin decompositions.
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. Dihedral group is one particular example of semidirect product of $\mathbb{Z}_n$ and $\mathbb{Z}_2$. 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 $\mathbb{Z}_N\rtimes_ϕ\mathbb{Z}_2$ when $N$ is odd with radical of $N$ being a Burnside number. As an application we also find the corresponding number of skew braces.
Let $L/K$ be a finite Galois extension of fields with group $Γ$. When $Γ$ is nilpotent, we show that the problem of enumerating all nilpotent Hopf-Galois structures on $L/K$ can be reduced to the corresponding problem for the Sylow subgroups of $Γ$. We use this to enumerate all nilpotent (resp. abelian) Hopf-Galois structures on a cyclic extension of arbitrary finite degree. When $Γ$ is abelian, we give conditions under which every abelian Hopf-Galois structure on $L/K$ has type $Γ$. We also give a criterion on $n$ such that \emph{every} Hopf-Galois structure on a cyclic extension of degree $n…
We study the question of the surjectivity of the Galois correspondence from subHopf algebras to subfields given by the Fundamental Theorem of Galois Theory for abelian Hopf Galois structures on a Galois extension of fields with Galois group G', a finite abelian p-group. Applying the connection between regular subgroups of the holomorph of a finite abelian p-group (G, +) and associative, commutative nilpotent algebra structures A on (G, +) of Caranti, et. al., we show that if A gives rise to a H-Hopf Galois structure on L/K, then the K-subHopf algebras of H correspond to the ideals of A. Among …
Let $H$ be a Hopf algebra, and $A,B$ be $H$-Galois extensions. We investigate the category ${}_A\mathcal{M}_B^H$ of relative Hopf bimodules, and the Morita equivalences between $A$ and $B$ induced by them.
Let H be a cocommutative faithfully flat Hopf quasigroup in a strict symmetric monoidal category with equalizers. In this paper we introduce the notion of (strong) Galois H-object and we prove that the set of isomorphism classes of (strong) Galois H-objects is a (group) monoid which coincides, in the Hopf algebra setting, with the Galois group of H-Galois objects introduced by Chase and Sweedler.
We introduce the concept of Hopf-Galois system, a reformulation of the notion of Galois extension of the base field for a Hopf algebra. The main feature of our definition is a generalization of the antipode of an ordinary Hopf algebra. The main application of Hopf-Galois systems is the construction of monoidal equivalences between comodule categories over Hopf algebras.
Let $K$ be a finite field extension of $\Q$ and let $N$ be a finite group with automorphism group $F=\Aut(N)$. R. Haggenmüller and B. Pareigis have shown that there is a bijection \[Θ: {\mathcal Gal}(K,F)\rightarrow {\mathcal Hopf}(K[N])\] from the collection of $F$-Galois extensions of $K$ to the collection of Hopf forms of the group ring $K[N]$. For $N=C_n$, $n\ge 1$, $C_p^m$, $p$ prime, $m\ge 1$, and $N=D_3,D_4,Q_8$, we show that $\Q[N]$ admits an absolutely semisimple Hopf form $H$ and find $L$ for which $Θ(L)=H$. Moreover, if $H$ is the Hopf algebra given by a Hopf-Galois structure on a G…
We associate two linear categories with two objects to a module over the subalgebra of coinvariants of a Hopf-Galois extension, and prove that they are isomorphic. The structure Theorem for cleft extensions, and the Militaru \cStefan lifting Theorem can be obtained using these isomorphisms.
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 $L \otimes_K H = LN$ where $N \cong (G, \star)$. Then there exists a skew left brace $(G, \star, \circ)$ where $(G, \circ) \cong Γ$. We show that there is a bijective…
Let $n$ be a squarefree natural number, and let $G$, $Γ$ be two groups of order $n$. We determine the number of Hopf-Galois structures of type $G$ admitted by a Galois extension of fields with Galois group isomorphic to $Γ$. We give some examples, including a full treatment of the case where $n$ is the product of three primes.
This article is made up with two parts. In the first part, we generalize to the case when objects are faithfully flat over the ground ring, the full equivalence between the notions of Hopf-Galois extension and Hopf-Galois system. In the second part of the article, we show that the Kashiwara algebras introduced by Kashiwara in his study of crystal bases are Hopf-Galois objects.
We investigate Hopf-Galois structures on a cyclic field extension $L/K$ of squarefree degree $n$. By a result of Greither and Pareigis, each such Hopf-Galois structure corresponds to a group of order $n$, whose isomorphism class we call the type of the Hopf-Galois structure. We show that every group of order $n$ can occur, and we determine the number of Hopf-Galois structures of each type. We then express the total number of Hopf-Galois structures on $L/K$ as a sum over factorisations of $n$ into three parts. As examples, we give closed expressions for the number of Hopf-Galois structures on a…
Let $H$ be a Hopf algebra. Ju and Cai introduced the notion of twisting of an $H$-module coalgebra. In this note, we study the relationship between twistings, crossed coproducts and Hopf-Galois coextensions. In particular, we show that a twisting of an $H$-Galois coextension remains $H$-Galois if the twisting is invertible.
Let $H$ be a Hopf algebra, and $A$ an $H$-Galois extension. We investigate $H$-Morita autoequivalences of $A$, introduce the concept of $H$-Picard group, and we establish an exact sequence linking the $H$-Picard group of $A$ and the Picard group of $A^{{\rm co}H}$.
Let $L/K$ be a finite Galois extension of fields with group $Γ$. Associated to each Hopf-Galois structure on $L/K$ is a group $G$ of the same order as the Galois group $Γ$. The type of the Hopf-Galois structure is by definition the isomorphism type of $G$. We investigate the extent to which general properties of either of the groups $Γ$ and $G$ constrain those of the other. Specifically, we show that if $G$ is nilpotent then $Γ$ is soluble, and that if $Γ$ is abelian then $G$ is soluble. The proof of the latter result depends on the classification of finite simple groups. In contrast to these …
Hopf-Galois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group $G$ are the Hopf-Galois extensions with respect to the dual of the group algebra of $G$. Rognes recently defined an analogous notion of Hopf-Galois extensions in the category of structured ring spectra, motivated by the fundamental example of the unit map from the sphere spectrum to $MU$. This article introduces a theory of homotopic Hopf-Galois extensions in a monoidal category with compatible model category structure that…
We introduce partial (co)actions of a Hopf algebra $H$ on an algebra. To this end, we introduce first the notion of lax coring, generalizing Wisbauer's notion of weak coring. We also have the dual notion of lax ring. Several duality results are given, and we develop Galois theory for partial $H$-comodule algebras.
We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.
Let $H$ be a finite dimensional Hopf algebra, and let $A$ be a left $H$-module algebra. Motivated by the study of the isolated singularities of $A^H$ and the endomorphism ring $\mathrm{End}_{A^H}(A)$, we introduce the concept of Hopf dense Galois extensions in this paper. Hopf dense Galois extensions yield certain equivalences between the quotient categories over $A$ and $A^H$. A special class of Hopf dense Galois extensions consits of the so-called densely group graded algebras, which are weaker versions of strongly graded algebras. A weaker version of Dade's Theorem holds for densely group g…
Descent theory for linear categories is developed. Given a linear category as an extension of a diagonal category, we introduce descent data, and the category of descent data is isomorphic to the category of representations of the diagonal category, if some flatness assumptions are satisfied. Then Hopf-Galois descent theory for linear Hopf categories, the Hopf algebra version of a linear category, is developed. This leads to the notion of Hopf-Galois category extension. We have a dual theory, where actions by dual linear Hopf categories on linear categories are considered. Hopf-Galois category…
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 a generalized ambiskew ring, based on a suitable Hopf algebra H(R). Several examples are examined, including the Hopf-Galois objects over Uq(sl2).
Let $(H,α)$ be a monoidal Hom-Hopf algebra, and $(A,β)$ a Hom-algebra. In this paper we will introduce the crossed product $(A\#_σH,β\otimesα)$, which is a Hom-algebra. Then we will introduce the notions of cleft extensions and Galois extensions respectively, and prove that a crossed product is equivalent to a cleft extension and a cleft extension is equivalent to a Galois extension with normal bases property.
We investigate criteria for algebra extensions that are of Galois type with respect to the coaction of a Hopf algebra or, more generally, a one-sided quotient of a Hopf algebra, or with respect to an entwining. We study the module- and comodule-theoretical properties of such extensions.
We introduce the concept of homotopy equivalence for Hopf Galois extensions and make a systematic study of it. As an application we determine all H-Galois extensions up to homotopy equivalence in the case when H is a Drinfeld-Jimbo quantum group.
By using the action of certain Galois groups on complex irreducible characters and conjugacy classes, we define the Galois characters and Galois classes. We will introduce a set of Galois characters, called Galois irreducible characters, and we show that each Galois character is a non-negative linear combination of the Galois irreducible characters. It is shown that whenever the complex characters of the groups of a tower produce a positive self-dual Hopf algebra (PSH), Galois characters of the groups of the tower also produce a PSH. Then we will classify the Galois characters and Galois class…
Let $S$ be the left $R$-bialgebroid of a depth two extension with centralizer $R$ as defined in math.QA/0108067. We show that the left endomorphism ring of depth two extension, not necessarily balanced, is a left $S$-Galois extension of $A^{\rm op}$. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We find a class of examples of the alternative Hopf algebroids in math.QA/0302325. We also characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are…
We define comodule algebras and Galois extensions for actions of bialgebroids. Using just module conditions we characterize the Frobenius extensions that are Galois as depth two and right balanced extensions. As a corollary, we obtain characterizations of certain weak and ordinary Hopf-Galois extensions without reference to action in the hypothesis.
A coaction of a Hopf algebra on a unital algebra is called homogeneous if the algebra of coinvariants equals the ground field. A coaction of a Hopf algebra on a (not necessarily unital) algebra is called Galois, or principal, or free, if the canonical map, also known as the Galois map, is bijective. In this paper, we establish a duality between a particular class of homogeneous coactions, up to equivariant Morita equivalence, and Galois coactions, up to isomorphism.
A pseudo-Galois extension is shown to be a depth two extension. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups, or more generally involutive Hopf algebras, and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observe that this Hopf algebroid is non-trivially isomorphic to a Connes-Moscovici Hopf algebroid, which raises interesting comparative questions.
We describe the Galois objects and biGalois groups of monomial nonsemisimple Hopf algebras. The main feature of our description is the use of modified versions of the second cohomology group of the grouplike elements. These computations generalize the previous ones of Masuoka and Schauenburg for the Taft algebras.
A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,μ)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the o…
Let $L/K$ be a Galois extension of fields with Galois group $G$, an elementary abelian $p$-group of rank $n$ for $p$ an odd prime. It is known that nilpotent $\mathbb{F}_p$-algebra structures $A$ on $G$ yield regular subgroups of the holomorph of $G$, hence Hopf Galois structures on $L/K$. In this paper we illustrate the richness of Hopf Galois structures on $L/K$ by examining the case where $A$ is abelian of dimension $n$ where the dimension of $A^2 = 1$. We determine the number of Hopf Galois structures that arise in these cases, describe those structures explicitly, and estimate the extent …
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 charac…
We define a bi-skew brace to be a set $G$ with two group operations $\star$ and $\circ$ so that $(G, \circ, \star)$ is a skew brace with additive group $(G, \star)$ and also with additive group $(G, \circ)$. If $G$ is a skew brace, then $G$ corresponds to a Hopf Galois structure of type $(G, \star)$ on any Galois extension of fields with Galois group isomorphic to $(G, \circ)$. If $G$ is a bi-skew brace, then $G$ also corresponds to a Hopf Galois structure of type $(G, \circ)$ on a Galois extension of fields with Galois group isomorphic to $(G, \star)$. Many non-trivial examples exist. One sou…
It is shown that a Hopf algebra over a field admitting a Galois extension separable over its subalgebra of coinvariants is of finite dimension. This answers in the affirmative a question posed by Beattie et al. in [{\it Proc. Amer. Math. Soc.} 128 No. 11 (2000), 3201-3203].
For $p$ a prime and $a\in\mathbb{Q}$, where $a$ is not a $p^n$-th power of any rational number, the extension $\mathbb{Q}(w_n)/\mathbb{Q}$ where $w_n=\root p^n \of a$ is separable but non-normal. The Hopf-Galois theory for separable extensions was determined by Greither and Pareigis, and the specific classification for radical extensions such as these by the author. In this work we extend this theory to a certain class of profinite extensions, namely those formed from the union of these $\mathbb{Q}(w_n)$. We construct a 'profinite' Hopf algebra which acts, and show that it satisfies a generali…
In this article we revisit the theory of homotopic Hopf-Galois extensions introduced in arXiv:0902.3393v2 [math.AT], in light of the homotopical Morita theory of comodules established in arXiv:1411.6517 [math.AT]. We generalize the theory to a relative framework, which we believe is new even in the classical context and which is essential for treating the Hopf-Galois correspondence in forthcoming work of the second author and Karpova. We study in detail homotopic Hopf-Galois extensions of differential graded algebras over a commutative ring, for which we establish a descent-type characterizati…
In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew polynomial rings, PBW extensions and skew PBW extensions, etc.). We collect and systematize questions, problems, properties and recent advances in both theories by explicitly developing examples and doing calculations that are usually omitted in the literature. In particular, for Hopf Galois extensions we consider approaches from the point of view of quantum torsors …
Let $L/K$ be a $G$-Galois extension of fields with an $H$-Hopf Galois structure of type $N$. We study the ratio $GC(G, N)$, which is the number of intermediate fields $E$ with $K \subseteq E \subseteq L$ that are in the image of the Galois correspondence for the $H$-Hopf Galois structure on $L/K$, divided by the number of intermediate fields. By Galois descent, $L \otimes_K H = LN$ where $N$ is a $G$-invariant regular subgroup of $\mathrm{Perm}(G)$, and then $GC(G, N)$ is the number of $G$-invariant subgroups of $N$, divided by the number of subgroups of $G$. We look at the Galois corresponden…
We reduce certain proofs in math.RA/0108067, math.RA/0408155, and math.QA/0409589 to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a left $T$-Galois extension for some right finite projective left bialgebroid over some algebra $R$ if and only if it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left…
$\DeclareMathOperator{\Aut}{Aut}$Let $p, q$ be distinct primes, with $p > 2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are cyclic. This is equivalent to classifying the skew braces $(G, \cdot, \circ)$. Furthermore, we prove that if $G$ and $Γ$ are groups of order $p^{2} q$ with non-is…
