Galois Theory for H-extensions and H-coextensions

TL;DR

This paper develops a Galois theory framework for H-extensions and H-coextensions, establishing correspondences between subalgebras, quotients, and coextensions in Hopf algebra contexts, with applications to finite-dimensional cases.

Contribution

It introduces a unified Galois correspondence for H-extensions and coextensions, including new conditions for bijections and closedness in crossed product scenarios.

Findings

Galois correspondence between subalgebras and quotients of H

Closedness of Q-Galois subextensions and H-Galois coextensions

Bijection between right ideal coideals and left coideal subalgebras in finite-dimensional H

Abstract

We show that there exists a Galois correspondence between subalgebras of an H-comodule algebra A over a base ring R and generalised quotients of a Hopf algebra H. We also show that Q-Galois subextensions are closed elements of the constructed Galois connection. Then we consider the theory of coextensions of H-module coalgebras. We construct Galois theory for them and we prove that H-Galois coextensions are closed. We apply the obtained results to the Hopf algebra itself and we show a simple proof that there is a bijection correspondence between right ideal coideals of H and its left coideal subalgebras when H is finite dimensional. Furthermore we formulate necessary and sufficient conditions when the Galois correspondence is a bijection for arbitrary Hopf algebras. We also present new conditions for closedness of subalgebras and generalised quotients when A is a crossed product.