Generic Hopf Galois extensions

In previous joint work with Eli Aljadeff we attached a generic Hopf Galois extension A H to each twisted algebra H obtained from a Hopf algebra H by twisting its product with the help of a cocycle α. The algebra A H is a flat deformation of H over a “big” central subalgebra B H and can be viewed as the noncommutative analogue of a versal torsor in the sense of Serre. After surveying the results on A H obtained with Aljadeff, we establish three new results: we present a systematic method to construct elements of the commutative algebra B H , we show that a certain important integrality condition is satisfied by all finite-dimensional Hopf algebras generated by grouplike and skew-primitive elements, and we compute B H in the case where H is the Hopf algebra of a cyclic group. Introduction In this paper we deal with associative algebras H obtained from a Hopf algebra H by twisting its product by a cocycle α. This class of algebras, which for simplicity we call twisted algebras, coincides with the class of so-called cleft Hopf Galois extensions of the ground field; classical Galois extensions and strongly group-graded algebras belong to this class. As has been stressed many times (see, e.g., [22]), Hopf Galois extensions can be viewed as noncommutative analogues of principal fiber bundles (also known as G-torsors), for which the rôle of the structural group is played by a Hopf algebra. Hopf Galois extensions abound in the world of quantum groups and of noncommutative geometry. The problem of constructing systematically Hopf Galois extensions of a given algebra for a given Hopf algebra and of classifying them up to isomorphism has been addressed in a number of papers over the last fifteen years; let us mention [4, 5, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21]. This list is far from being exhaustive, but gives a pretty good idea of the activity on this subject. A new approach to this problem was recently considered in [2]; this approach mixes commutative algebra with techniques from noncommutative algebra such as polynomial identities. In particular, in that paper Eli Aljadeff and the author 2000 Mathematics Subject Classification. Primary (16W30, 16S35, 16R50) Secondary (13B05, 13B22, 16E99, 58B32, 58B34, 81R50).