Generic Hopf Galois extensions

TL;DR

This paper studies generic Hopf Galois extensions, presenting new methods for constructing their base algebras, proving integrality conditions for certain Hopf algebras, and explicitly computing examples for cyclic groups.

Contribution

It introduces a systematic way to construct elements of the base algebra, proves an integrality condition for specific Hopf algebras, and computes the base algebra for cyclic group Hopf algebras.

Findings

A method to construct elements of B(H,c)

Finite-dimensional Hopf algebras generated by grouplike and skew-primitive elements satisfy an integrality condition

B(H,c) computed explicitly for cyclic group Hopf algebras

Abstract

In previous joint work with Eli Aljadeff we attached a generic Hopf Galois extension A(H,c) to each twisted algebra H(c) obtained from a Hopf algebra H by twisting its product with the help of a cocycle c. The algebra A(H,c) is a flat deformation of H(c) over a "big" central subalgebra B(H,c) and can be viewed as the noncommutative analogue of a versal torsor in the sense of Serre. After surveying the results on A(H,c) obtained with Aljadeff, we establish three new results: we present a systematic method to construct elements of the commutative algebra B(H,c), 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,c) in the case where H is the Hopf algebra of a cyclic group.