Coalgebra Extensions and Algebra Coextensions of Galois Type

TL;DR

This paper generalizes the concept of Galois extensions to coalgebra and algebra contexts, establishing the existence and uniqueness of entwining structures compatible with their respective coactions and actions.

Contribution

It introduces a unified framework for coalgebra and algebra Galois extensions, proving the existence and uniqueness of compatible entwining maps.

Findings

Existence of unique entwining maps for coalgebra-Galois extensions.

Existence of unique entwining structures for algebra-Galois coextensions.

Generalization of Hopf-Galois extensions to broader coalgebra and algebra settings.

Abstract

The notion of a coalgebra-Galois extension is defined as a natural generalisation of a Hopf-Galois extension. It is shown that any coalgebra-Galois extension induces a unique entwining map $\psi$ compatible with the right coaction. For the dual notion of an algebra-Galois coextension it is also proven that there always exists a unique entwining structure compatible with the right action.