On the main theorem of separable hopf Galois theory
Author
Vela, M.; Crespo, T.; Rio, A.
Type of activity
Presentation of work at congresses
Name of edition
28è Journées Arithmétiques
Date of publication
2013
Presentation's date
2013-07-01
Abstract
Chase and Sweedler developed the concept of Hopf Galois extension $K/k$ for a $k$-Hopf algebra $H$. In particular, they proved the main theorem in its general form: the map from the set of sub-Hopf algebras of $H$ to the set of intermediate subfields of $K/k$ taking a sub-Hopf algebra of $H$ to its fixed field is injective and inclusion reversing. If this map is also surjective, we say that the main theorem holds in its strong form.
Greither and Pareigis considered the case of separable exte...
Chase and Sweedler developed the concept of Hopf Galois extension $K/k$ for a $k$-Hopf algebra $H$. In particular, they proved the main theorem in its general form: the map from the set of sub-Hopf algebras of $H$ to the set of intermediate subfields of $K/k$ taking a sub-Hopf algebra of $H$ to its fixed field is injective and inclusion reversing. If this map is also surjective, we say that the main theorem holds in its strong form.
Greither and Pareigis considered the case of separable extensions and showed how the Hopf Galois structures of these extensions can be described in group-theoretic terms. For the almost classically Galois extensions they proved that there exists a Hopf Galois structure such that the main theorem holds in its strong form.
We present a reformulation of the main theorem in terms of groups following the description of Greither and Pareigis. We provide some examples of Galois extensions, almost classically Galois extensions and not almost classically Galois Hopf Galois extensions and we describe the lattice of intermediate extensions corresponding to the lattice of sub-Hopf algebras, revealing the differences between not equivalent Hopf Galois structures.