Forms of Hopf Algebras and Galois Theory.
In: Banach Center Publications, Vol. 26: pp. 75-93
The theory of Hopf algebras is closely connected with various applications,
in particular to algebraic and formal groups. Although the
rst occurence of Hopf algebras was in algebraic topology, they are now
found in areas as remote as combinatorics and analysis. Their structure
has been studied in great detail and many of their properties are
well understood. We are interested in a systematic treatment of Hopf
algebras with the techniques of forms and descent.
The rst three paragraphs of this paper give a survey of the present
state of the theory of forms of Hopf algebras and of Hopf Galois theory
especially for separable extensions. It includes many illustrating examples
some of which cannot be found in detail in the literature. The last
two paragraphs are devoted to some new or partial results on the same
eld. There we formulate some of the open questions which should
be interesting objects for further study. We assume throughout most
of the paper that k is a base eld and do not touch upon the recent
beautiful results of Hopf Galois theory for rings of integers in algebraic
number elds as developed in [C1].