Forms of Hopf algebras and Galois theory

Abstract

The author reviews and extends some of his recent work on Hopfalgebras and Galois extension over fields. The first sectionreviews the Hopf algebra forms of group rings kG with\vert {\rm Aut}(G)\vert =2 of R. Haggenmuller and the author[\Cite{Haggenmuller86:Hopf:121--136}[87e:16026]Manuscripta Math.55 (1986), no. 2, 121 136; MR 87e:16026], a result which exploitsthe identification of the cohomology group H\sp 1(K/k,\underline{\rm Aut}(H)) as k forms of a Hopf algebra H andas forms of Galois extensions of k with Galois group{\rm Aut}(H). This result is extended in Section 4 to the casewhere H=kC\sb 5, {\rm Aut}(H)=C\sb 4, where various openquestions about Hopf algebra forms of group rings and theirrepresentation theory are also raised. After a review of HopfGalois extensions and principal homogeneous spaces in Section 2,the third section reviews the work of C. Greither and the author[\Cite{Greither87:Hopf:239--258}[88i:12006]J. Algebra 106 (1987),no. 1, 239 258; MR 88i:12006] on Hopf Galois structures onseparable field extensions, and in the final section the authorextends a result of the reviewer[\Cite{Childs89:Hopf:809--825}[90g:12003]Comm. Algebra 17 (1989),no. 4, 809 825; MR 90g:12003] to show that for a separable fieldextension K/k of prime degree p, with normal closure K',K/k is H Galois for some k Hopf algebra H if and only if{\rm Gal}(K'/k) is solvable, in which case H is the uniquek Hopf algebra making K/k Galois.\par {For the entirecollection see MR\Cite{Balcerzyk90:Topics:PWN---Polish}[93a:00023].}