Galois Theory of Hopf Galois Extensions

TL;DR

This paper develops a Galois theory framework for Hopf-Galois extensions, establishing a correspondence between subalgebras and quotients of Hopf algebras, and extends key results to finite and infinite dimensional cases.

Contribution

It introduces a Galois connection for Hopf-Galois extensions, generalizes known results, and characterizes closed elements as Galois extensions, advancing the theoretical understanding of Hopf algebra symmetries.

Findings

Established a Galois connection between subalgebras and quotients

Proved bijective correspondence for finite dimensional Hopf algebras

Characterized closed elements as Hopf-Galois extensions

Abstract

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 define Q-Galois extension are the closed elements of our Galois connection. We generalise important results of Hopf--Galois Theory of M. Masuoka and H.-J. Schneider by showing that there is a bijective correspondence between right ideals coideals and right coideal subalgebras of any finite dimensional Hopf algebra and we reformulate the still open problem in the general (i.e. infinite dimensional) case. For cleft extensions we characterise closed elements of the Galois connection as Hopf-Galois extensions. We describe the relation of our results to the work of F. van Oystaeyen, Y. Zhang and also to the results of P. Schauenburg on biGalois extensions.