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 Galois-theoretic flavour. In this paper, we mainly prove a criterion for an Ore extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, and introduce the conception of Poisson Hopf-Galois algebras, and establish the relationship between Poisson Hopf-Galois algebras and Poisson Hopf algebras. Moreover, we study Poisson Hopf-Galois structures on Poisson polynomial algebras, and mainly give a necessary and sufficient condition for the Poisson enveloping algebra of a Poisson Hopf-Galois algebra to be a Hopf-Galois algebra.
本文引入了Hopf-Galois扩张的伽罗瓦论，证明了H-余代数的子代数与霍普夫代数H的广义余代数之间存在Galois联络。此外，我们还证明了定义Q-Galois扩张的这些行列式Q是我们的Galois联络的闭元。推广了M. Masuoka和H. J. Schneider的结果，证明了任意有限维霍普夫代数的右理想、余理想和右余理想子代数之间存在双射对应关系，并在一般（即无限维）情况下重新表述了这一尚未解决的问题.对于分裂扩张，我们将Galois联络的闭元作为Hopf-Galois扩张。我们描述了我们的结果与F. van Oystaeyen，Y. Zhang和P. Schauenburg关于双伽罗瓦扩张的结果。
We survey some aspects of the theory of Hopf-Galois objects that may studied advantageously by using the language of cogroupoids. These are the notes for a series of lectures given at Cordoba University, may 2010. The lectures are part of the course ``Hopf-Galois theory'' by Sonia Natale.
We show that the matrix embeddings in Bratteli diagrams are iterated direct sums of Hopf-Galois extensions (quantum principle bundles) for certain abelian groups. The corresponding strong universal connections are computed. We show that $ M_{n}(\mathbb{C})$ is a trivial quantum principle bundle for the Hopf algebra $ \mathbb{C}[\mathbb{Z}_{n} \times \mathbb{Z}_{n}] $. We conclude with an application relating known calculi on groups to calculi on matrices.
To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then shown that this assignment establishes an equivalence between categories of Hopf heaps and Hopf-Galois co-objects.
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 $L(P,H)$. We show further that if $H$ is coquasitriangular then $L(P,H)$ has an antipode map $S$ obeying certain minimal axioms. Trivial quantum principal bundles or cleft Hopf Galois extensions with base $B$ are known to be cocycle cross products $B\#_σH$ for a cocycle-action pair ($\vartriangleright$,$σ$) and we look at these of a certain `associative type' where $ \vartriangleright$ is an actual action. In this case a…
We construct an invertible normalised 2 cocycle on the Ehresmann Schauenburg bialgebroid of a cleft Hopf Galois extension under the condition that the corresponding Hopf algebra is cocommutative and the image of the unital cocycle corresponding to this cleft Hopf Galois extension belongs to the centre of the coinvariant subalgebra. Moreover, we show that any Ehresmann Schauenburg bialgebroid of this kind is isomorphic to a 2-cocycle twist of the Ehresmann Schauenburg bialgebroid corresponding to a Hopf Galois extension without cocycle, where comodule algebra is an ordinary smash product of the…
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 …
