Hopf Galois Extensions Research Articles

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.

Differential Calculi on Quantum Principal Bundles Over Projective Bases

We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of covariant bimodules together with a morphism of sheaves -the differential- such that the Leibniz rule and surjectivity hold locally. The main class of examples is given by covariant calculi over quantum flag manifolds, which we provide via an explicit Ore extension construction. In a second step we introduce principal covariant calculi by requiring a local compatibility of the calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms of exact sequences. In this case Hopf–Galois extensions of algebras lift to Hopf–Galois extensions of exterior algebras with compatible differentials. In particular, the examples of principal (covariant) calculi on the quantum principal bundles Oq(SL2(C))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_q(\ extrm{SL}_2(\\mathbb {C}))$$\\end{document} and Oq(GL2(C))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_q(\ extrm{GL}_2(\\mathbb {C}))$$\\end{document} over the projective space P1(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{P}^1(\\mathbb {C})$$\\end{document} are discussed in detail.

Hopf-Galois extensions and twisted Hopf algebroids

We propose a new notion of antipode S for a left Hopf algebroid which does not assume antimultiplicativity. We also show that if L is a left Hopf algebroid then so is its cotwist Lς as an extension of a previous bialgebroid Drinfeld cotwist theory. We show that the Ehresmann-Schauenburg bialgebroid of a quantum principal bundle P or Hopf-Galois extension with structure quantum group H is in fact a left Hopf algebroid L(P,H) and that it has an antipode S at least if H is coquasitriangular or if the bundle is trivial so that P=B#σH is a cleft extension and the cocycle-action (▹,σ) is of associative type. We also show in the latter case that L(B#σH,H)=L(B#H,H)σ˜ for a Hopf algebroid cotwist ς=σ˜. Thus, introducing a σ of associative type appears at the Hopf algebroid level as a Drinfeld cotwist. We view the affine quantum group Uq(sl2)ˆ and the quantum Weyl group of uq(sl2) as examples of associative type.

Gorenstein Flat Modules of Hopf-Galois Extensions

Let A/B be a right H-Galois extension over a semisimple Hopf algebra H. The purpose of this paper is to give the relationship of Gorenstein flat dimensions between the algebra A and its subalgebra B, and obtain that the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A is no more than that of B. Then the problem of preserving property of Gorenstein flat precovers for the Hopf-Galois extension will be studied. Finally, more relations for the crossed products and smash products will be obtained as applications.

Hopf Differential Graded Galois Extensions

We introduce the concept of Hopf dg Galois extensions. For a finite dimensional semisimple Hopf algebra H and an H-module dg algebra R, we show that D(R#H)≅D(RH) is equivalent to that R/RH is a Hopf differential graded Galois extension. We present a weaker version of Hopf differential graded Galois extensions and show the relationships between Hopf differential graded Galois extensions and Hopf Galois extensions.

On the passage from finite braces to pre-Lie rings

Let p be a prime number. We show that there is a one-to-one correspondence between the set of strongly nilpotent braces and the set of nilpotent pre-Lie rings of cardinality pn, for sufficiently large p. Moreover, there is an injective mapping from the set of left nilpotent pre-Lie rings into the set of left nilpotent braces of cardinality pn for n+1<p. As an application, by using well known results about the correspondence between braces and Hopf-Galois extensions we use pre-Lie algebras to describe Hopf-Galois extensions.

Some interactions between Hopf Galois extensions and noncommutative rings

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 (also known as quantum heaps) and Hopf Galois systems, while for some families of noncommutative rings we present advances in the characterization of ring-theoretic and homological properties. Every developed topic is exemplified with abundant references to classic and current works, so this paper serves as a survey for those interested in either of the two theories. Throughout, interactions between both are presented.

Equivalences of (co)module algebra structures over Hopf algebras

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra A, there exists a unique universal Hopf algebra H together with an H-(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. We apply support equivalence in the study of the asymptotic behaviour of codimensions of H-identities and, in particular, to the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture, which was originally concerned with ordinary polynomial identities. As an example we prove this analogue for all unital H-module structures on the algebra $F[x]/(x^2)$ of dual numbers.

The gauge group of a noncommutative principal bundle and twist deformations

We study noncommutative principal bundles (Hopf–Galois extensions) in the context of coquasitriangular Hopf algebras and their monoidal category of comodule algebras. When the total space is quasi-commutative, and thus the base space subalgebra is central, we define the gauge group as the group of vertical automorphisms or equivalently as the group of equivariant algebra maps.We study Drinfeld twist (2-cocycle) deformations of Hopf–Galois extensions and show that the gauge group of the twisted extension is isomorphic to the gauge group of the initial extension. In particular, noncommutative principal bundles arising via twist deformation of commutative principal bundles have classical gauge group. We illustrate the theory with a few examples.

Skew braces of size pq

We construct all skew braces of size pq (where p > q are primes) by using Byott’s classification of Hopf–Galois extensions of the same degree. For there exists only one skew brace which is the trivial one. When we have skew braces, two of which are of cyclic type (so, contained in Rump’s classification) and 2q of non-abelian type.

Homotopical Morita theory for corings

A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent the ...

Homotopic Hopf–Galois extensions revisited

In this article we revisit the theory of homotopic Hopf–Galois extensions introduced in [9], in light of the homotopical Morita theory of comodules established in [3].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 [19]. We study in detail homotopic Hopf–Galois extensions of differential graded algebras over a commutative ring, for which we establish a descent-type characterization analogous to the one Rognes provided in the context of ring spectra [26]. An interesting feature in the differential graded setting is the close relationship between homotopic Hopf–Galois theory and Koszul duality theory. We show that nice enough principal fibrations of simplicial sets give rise to homotopic Hopf–Galois extensions in the differential graded setting, for which this Koszul duality has a familiar form.

On skew braces (with an appendix by N. Byott and L. Vendramin)

Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang–Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf–Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.

Note on the Hochschild cohomology of Hopf–Galois extension

We generalize [4, Theorem 4.3] to the case of Hopf–Galois extension, by introducing the cotensor product of a comodule algebra and its opposite algebra, and then give some applications.

On representation theory of total (co)integrals

In this paper, we show that total integrals and cointegrals are new sources of stable anti-Yetter–Drinfeld modules. We explicitly show that how special types of total (co)integrals can be used to provide both (stable) anti Yetter–Drinfeld and Yetter–Drinfeld modules. We use these modules to classify total (co)integrals and (cleft) Hopf Galois (co)extensions of the Connes–Moscovici Hopf algebra, and some examples of universal enveloping algebra and polynomial algebra.

Noncommutative Principal Bundles Through Twist Deformation

We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new Hopf-Galois extensions as twists of Hopf-Galois extensions. A sheaf approach is also considered, and examples presented.

Fixed Point Algebras for Easy Quantum Groups

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their $K$-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the $K$-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group $S_n^+$, the free orthogonal quantum group $O_n^+$ and the quantum reflection groups $H_n^{s+}$. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions.

Deformation quantization of principal bundles

We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles and, more in general, to the deformation of Hopf–Galois extensions. First, we twist deform the structure group in a quantum group, and this leads to a deformation of the fibers of the principal bundle. Next, we twist deform a subgroup of the group of automorphisms of the principal bundle, and this leads to a noncommutative base space. Considering both deformations, we obtain noncommutative principal bundles with noncommutative fiber and base space as well.

Finite Dimensional Hopf Actions on Central Division Algebras

Let $\mathbb{k}$ be an algebraically closed field of characteristic zero. Let $D$ be a division algebra of degree $d$ over its center $Z(D)$. Assume that $\mathbb{k}\subset Z(D)$. We show that a finite group $G$ faithfully grades $D$ if and only if $G$ contains a normal abelian subgroup of index dividing $d$. We also prove that if a finite dimensional Hopf algebra coacts on $D$ defining a Hopf-Galois extension, then its PI degree is at most $d^2$. Finally, we construct Hopf-Galois actions on division algebras of twisted group algebras attached to bijective cocycles.

Counterexample to a conjecture about braces

We find an example of a finite solvable group (in fact, a finite p-group) in which is not possible to define a left brace structure or, equivalently, which is not an IYB group. This answers a question posed by Cedó, Jespers and del Río related to the Yang–Baxter equation. Our argument is an improvement of an argument of Rump, using results about Hopf–Galois extensions and LSA structures. We explain explicitly the relation between these two areas of mathematics and brace theory, hoping that it will be useful in the future.