Related papers: Some interactions between Hopf Galois extensions a…
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…
We study Hopf Galois extensions of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we introduce (skew-)regular comodules and generalize the structure theorem for relative Hopf modules. Also, we show that…
We review Hopf-Galois extensions, in particular faithfully flat ones, accepted to be the noncommutative algebraic dual of a principal bundle. We also make a short digression into how quantum groups relate to Hopf-Galois extensions. Several…
This paper is a written form of a talk. It gives a review of various notions of Galois (and in particular cleft) extensions. Extensions by coalgebras,bialgebras and Hopf algebras (over a commutative base ring) and by corings,bialgebroids…
Hopf algebroids are generalization of Hopf algebras over non-commutative base rings. It consists of a left- and a right-bialgebroid structure related by a map called the antipode. However, if the base ring of a Hopf algebroid is commutative…
Coalgebra-Galois extensions generalise Hopf-Galois extensions, which can be viewed as non-commutative torsors. In this paper it is analysed when a coalgebra-Galois extension is a separable, split, or strongly separable extension.
As is known to all, Hopf-Galois objects have a significant research value for analyzing tensor categories of comodules and classification questions of pointed Hopf algebras, and are natural generalizations of Hopf algebras with a…
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…
Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable…
The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid $H= (H_L,H_R)$) is cleft if and only if it is $H_R$-Galois and has a…
We study the push-forward of Hopf--Galois extensions as the algebraic counterpart of the pullback of principal bundles. We apply the theory of twisted tensor product algebras to endow covariant extensions of modules along a map $\mathsf{F}$…
We introduce Galois Theory for Hopf-Galois Extensions proving existence of a Galois connection between subalgebras of an H-comodule algebra and generalised quotients of the Hopf algebra H. Moreover, we show that these quotients Q which…
Hopf crossed products, or in other words, cleft comodule algebras form a special but important class in Hopf-Galois extensions. To discuss this interesting subject, we will start with the more familiar group crossed products, and then see…
We understand quantum principal bundle as faithfully flat Hopf--Galois extensions, with a structure Hopf algebra coacting on a total space algebra and with base algebra given by the coinvariant elements. To endow such bundles with a…
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…
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…
We study Hopf-Galois extensions with central invariants for a finite dimensional Hopf algebra. We collect general facts about them and discuss some examples arising in the study of restricted Lie algebras and quantum groups at roots of…
Cyclic cohomology has been recently adapted to the treatment of Hopf symmetry in noncommutative geometry. The resulting theory of characteristic classes for Hopf algebras and their actions on algebras allows to expand the range of…
We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We…
A Hopf Galois structure on a finite field extension $L/K$ is a pair $(\mathcal{H},\mu)$, where $\mathcal{H}$ is a finite cocommutative $K$-Hopf algebra and $\mu$ a Hopf action. In this paper, we present several results on Hopf Galois…