Isabel Martin-Lyons (University of Keele) – The Hopf-Galois correspondence and skew bracoids

Hopf-Galois theory provides an analogue to Galois theory for non-Galois extensions, of particular interest are separable but not necessarily normal extensions of fields. A Hopf-Galois structure consists of a cocommutative Hopf algebra over the base field and an action of this on the top field satisfying various conditions, this functions much like a Galois group though an extension may admit multiple distinct Hopf-Galois structures or none at all. As in classical Galois theory, we have a notion of the Hopf-Galois correspondence; the natural fix map from the Hopf sub-algebras of such a structure to the intermediate fields of the extension is always injective and inclusion reversing but, unlike for the Galois correspondence, it is not in general surjective. Much work has been done to determine when the Hopf-Galois correspondence is surjective and the skew brace has proved to be a fruitful tool in the Galois case. We introduce the skew bracoid, a generalisation of the skew brace that corresponds to Hopf-Galois extensions on separable extensions of fields and comprises two groups connected by a compatible transitive action. We then outline how the skew bracoid may be of use in the study of the Hopf-Galois correspondence in the separable case, providing some preliminary results.