Hopf Galois Extensions Research Articles

On the Galois correspondence theorem in separable Hopf Galois theory

In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure.

The Hopf Galois Property in Subfield Lattices

Let K/k be a finite separable extension, n its degree and its Galois closure. For n ≤ 5, Greither and Pareigis show that all Hopf Galois extensions are either Galois or almost classically Galois and they determine the Hopf Galois character of K/k according to the Galois group (or the degree) of . In this paper we study the case n = 6, and intermediate extensions F/k such that , for degrees n = 4, 5, 6. We present an example of a non almost classically Galois Hopf Galois extension of ℚ of the smallest possible degree and new examples of Hopf Galois extensions. In the last section we prove a transitivity property of the Hopf Galois condition.

Some remarks on δ-Koszul algebras

Let A be a δ-Koszul algebra, and let Ƙδ(A) and L(A) denote the categories of δ-Koszul modules and modules with linear presentations. Some necessary and sufficient conditions for Ƙδ(A) = L(A) are given. Set $$\begin{array}{*{20}c} {E(A): = \mathop \oplus \limits_{i \geqslant 0} Ext_A^i (A_0 ,A_0 )} & {and} & {\mathcal{B}(A): = \sup \left\{ {\left. {i \in \mathbb{N}} \right|Ext_A^i (A_0 ,A_0 ) \cap V \ne 0} \right\},} \\ \end{array}$$ , where V is a minimal graded generating space of E(A). In the present paper, we prove that {B(A)| A is δ - Koszul} = ℕ. Finally, the Koszulity of the graded Hopf Galois extension of δ-Koszul algebras is studied.

Hopf–Galois extensions for monoidal Hom-Hopf algebras

Hopf–Galois extensions for monoidal Hom-Hopf algebras are investigated. As the main result, Schneider’s affineness theorem in the case of monoidal Hom-Hopf algebras is shown in terms of total integrals and Hopf–Galois extensions. In addition, we obtain an

A 4-Sphere With Noncentral Radius and its Instanton Sheaf

We build an SU(2)-Hopf bundle over a quantum toric four-sphere whose radius is noncentral. The construction is carried out using local methods in terms of sheaves of Hopf–Galois extensions. The associated instanton bundle is presented and endowed with a connection with anti-self-dual curvature.

Line bundles and the Thom construction in noncommutative geometry

The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we construct \mathbb{Z} - and \mathbb{N} -graded algebras, the \mathbb{Z} -graded algebra being a Hopf–Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the \mathbb{N} -graded algebra. In this case, we carry out the associated circle bundle and Thom constructions. Starting with a C*-algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C*-algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi and Nomizu.

On the Smoothness of the Noncommutative Pillow and Quantum Teardrops

Recent results by Kr\"ahmer [Israel J. Math. 189 (2012), 237-266, arXiv:0806.0267] on smoothness of Hopf-Galois extensions and by Liu [arXiv:1304.7117] on smoothness of generalized Weyl algebras are used to prove that the coordinate algebras of the noncommutative pillow orbifold [Internat. J. Math. 2 (1991), 139-166], quantum teardrops ${\mathcal O}({\mathbb W}{\mathbb P}_q(1,l))$ [Comm. Math. Phys. 316 (2012), 151-170, arXiv:1107.1417], quantum lens spaces ${\mathcal O}(L_q(l;1,l))$ [Pacific J. Math. 211 (2003), 249-263], the quantum Seifert manifold ${\mathcal O}(\Sigma_q^3)$ [J. Geom. Phys. 62 (2012), 1097-1107, arXiv:1105.5897], quantum real weighted projective planes ${\mathcal O}({\mathbb R}{\mathbb P}_q^2(l;\pm))$ [PoS Proc. Sci. (2012), PoS(CORFU2011), 055, 10 pages, arXiv:1203.6801] and quantum Seifert lens spaces ${\mathcal O}(\Sigma_q^3(l;-))$ [Axioms 1 (2012), 201-225, arXiv:1207.2313] are homologically smooth in the sense that as their own bimodules they admit finitely generated projective resolutions of finite length.

Categorical duality for Yetter-Drinfeld algebras

We study tensor structures on (\operatorname{Rep} G) -module categories defined by actions of a compact quantum group G on unital C ^* -algebras. We show that having a tensor product which defines the module structure is equivalent to enriching the action of G to the structure of a braided-commutative Yetter–Drinfeld algebra. This shows that the category of braided-commutative Yetter–Drinfeld G -C ^* -algebras is equivalent to the category of generating unitary tensor functors from \operatorname{Rep} G into C ^* -tensor categories. To illustrate this equivalence, we discuss coideals of quotient type in C(G) , Hopf–Galois extensions and noncommutative Poisson boundaries.

The homotopy theory of coalgebras over a comonad

Let 𝕂 be a comonad on a model category M. We provide conditions under which the associated category M𝕂 of 𝕂-coalgebras admits a model category structure such that the forgetful functor M𝕂→M creates both cofibrations and weak equivalences. We provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of Hopf–Galois extensions. These examples are specific instances of the following categories of comodules over a coring (co-ring). For any semihereditary commutative ring R, let A be a dg R-algebra that is homologically simply connected. Let V be an A-coring that is semifree as a left A-module on a degreewise R-free, homologically simply connected graded module of finite type. We show that there is a model category structure on the category MA of right A-modules satisfying the conditions of our existence theorem with respect to the comonad−⊗AV and conclude that the category MAV of V-comodules in MA admits a model category structure of the desired type. Finally, under extra conditions on R, A and V, we describe fibrant replacements in MAV in terms of a generalized cobar construction.

Equivariant Hopf Galois extensions and Hopf cyclic cohomology

We define the notion of equivariant \times -Hopf Galois extension and apply it as a functor between the categories of stable anti-Yetter–Drinfeld (SAYD) modules of the \times -Hopf algebras involved in the extension. This generalizes the result of Jara–Ştefan and Böhm–Ştefan on associating a SAYD modules to any ordinary Hopf Galois extension.

Homological dimension of weak Hopf–Galois extensions

Let H be a weak Hopf algebra, A a right weak H-comodule algebra and B the subalgebra of the H-coinvariant elements of A. Let A/B be a right weak H-Galois extension. We prove that A/B is a separable extension if H is semisimple. Using this, we show that the global dimension and weak dimension of A are less than those of B. As an application, we obtain Maschke-type theorems for weak Hopf–Galois extensions and weak smash products.

Quantum principal bundles over quantum real projective spaces

Two hierarchies of quantum principal bundles over quantum real projective spaces are constructed. One hierarchy contains bundles with U(1) as a structure group, the other has the quantum group SUq(2) as a fibre. Both hierarchies are obtained by the process of prolongation from bundles with the cyclic group of order 2 as a fibre. The triviality or otherwise of these bundles is determined by using a general criterion for a prolongation of a comodule algebra to be a cleft Hopf–Galois extension.

Projective Modules Over Frobenius Algebras and Hopf Comodule Algebras

This note presents some results on projective modules and the Grothendieck groups K 0 and G 0 for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions.

Representations of the category of modules over pointed Hopf algebras over 𝕊3and 𝕊4

We classify exact indecomposable module categories over the representation category of all non-trivial Hopf algebras with coradi- cal S3 and S4. As a byproduct, we compute all its Hopf-Galois extensions and we show that these Hopf algebras are cocycle deformations of their graded versions.

The structure theorem of endomorphism algebras for weak Doi-Hopf modules

The paper is concerned with endomorphism algebras for weak Doi-Hopf modules. Under the condition “weak Hopf-Galois extensions”, we present the structure theorem of endomorphism algebras for weak Doi-Hopf modules, which extends Theorem 3.2 given by Schneider in [1]. As applications of the structure theorem, we obtain the Kreimer-Takeuchi theorem (see Theorem 1.7 in [2]) and the Nikshych duality theorem (see Theorem 3.3 in [3]) in the case of weak Hopf algebras, respectively.

Non-commutative integral forms and twisted multi-derivations

Non-commutative connections of the second type or hom-connections and associated integral forms are studied as generalisations of right connections of Manin. First, it is proven that the existence of hom-connections with respect to the universal differential graded algebra is tantamount to the injectivity, and that every injective module admits a hom-connection with respect to any differential graded algebra. The bulk of the article is devoted to describing a method of constructing hom-connections from twisted multi-derivations . The notion of a free twisted multi-derivation is introduced and the induced first order differential calculus is described. It is shown that any free twisted multi-derivation on an algebra A induces a unique hom-connection on A (with respect to the induced differential calculus Ω^1(A) ) that vanishes on the dual basis of Ω^1(A) . To any flat hom-connection ∇ on A one associates a chain complex, termed a complex of integral forms on A . The canonical cokernel morphism to the zeroth homology space is called a ∇ - integral . Examples of free twisted multi-derivations, hom-connections and corresponding integral forms are provided by covariant calculi on Hopf algebras (quantum groups). The example of a flat hom-connection within the 3D left-covariant differential calculus on the quantum group \mathcal{O}_q(\mathrm{SL}(2)) is described in full detail. A descent of hom-connections to the base algebra of a faithfully flat Hopf–Galois extension or a principal comodule algebra is studied. As an example, a hom-connection on the standard quantum Podle’s sphere \mathcal{O}_q(\mathrm{SL}(2)) is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the ∇ -integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.

Coactions on Hochschild homology of Hopf–Galois extensions and their coinvariants

Let B ⊆ A be an H -Galois extension, where H is a Hopf algebra over a field K . If M is a Hopf bimodule then HH ∗ ( A , M ) , the Hochschild homology of A with coefficients in M , is a right comodule over the coalgebra C H = H / [ H , H ] . Given an injective left C H -comodule V , our aim is to understand the relationship between HH ∗ ( A , M ) □ C H V and HH ∗ ( B , M □ C H V ) . The roots of this problem can be found in Lorenz (1994) [15], where HH ∗ ( A , A ) G and HH ∗ ( B , B ) are shown to be isomorphic for any centrally G -Galois extension. To approach the above mentioned problem, in the case when A is a faithfully flat B -module and H satisfies some technical conditions, we construct a spectral sequence Tor p R H ( K , HH q ( B , M □ C H V ) ) ⟹ HH p + q ( A , M ) □ C H V , where R H denotes the subalgebra of cocommutative elements in H . We also find conditions on H such that the edge maps of the above spectral sequence yield isomorphisms K ⊗ R H HH ∗ ( B , M □ C H V ) ≅ HH ∗ ( A , M ) □ C H V . In the last part of the paper we define centrally Hopf–Galois extensions and we show that for such an extension B ⊆ A , the R H -action on HH ∗ ( B , M □ C H V ) is trivial. As an application, we compute the subspace of H -coinvariant elements in HH ∗ ( A , M ) . A similar result is derived for HC ∗ ( A ) , the cyclic homology of A .

Hopf–Galois extensions and an exact sequence for H-Picard groups

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 co H .

A characterization of projective weak Galois extensions

It is a well-known fact that in the definition of Hopf-Galois extensions, the bijectivity of the canonical morphism is an essential part. In this paper we obtain a criterion under which the surjectivity of the canonical morphism implies the bijectivity in a weak context, generalizing recent results of Schauenburg and Schneider, and Brzezinski, Turner and Wrightson.

On the Cartan Map for Crossed Products and Hopf–Galois Extensions

We study certain aspects of the algebraic K-theory of Hopf–Galois extensions. We show that the Cartan map from K-theory to G-theory of such an extension is a rational isomorphism, provided the ring of coinvariants is regular, the Hopf algebra is finite dimensional and its Cartan map is injective in degree zero. This covers the case of a crossed product of a regular ring with a finite group and has an application to the study of Iwasawa modules.