Hopf-Galois structures on ambiskew polynomial rings

TL;DR

This paper characterizes when Hopf-Galois algebra structures extend to generalized ambiskew rings, showing the resulting Hopf algebra retains a similar structure, with applications to quantum groups like Uq(sl2).

Contribution

It provides necessary and sufficient conditions for extending Hopf-Galois structures to ambiskew rings, revealing the extended Hopf algebra remains a generalized ambiskew ring.

Findings

Extension conditions are explicitly characterized.

The extended Hopf algebra is also a generalized ambiskew ring.

Applications include Hopf-Galois objects over Uq(sl2).

Abstract

We provide necessary and sufficient conditions to extend the Hopf-Galois algebra structure on an algebra R to a generalized ambiskew ring based on R, in a way such that the added variables for the extension are skew-primitive in an appropriate sense. We show that the associated Hopf algebra is again a a generalized ambiskew ring, based on a suitable Hopf algebra H(R). Several examples are examined, including the Hopf-Galois objects over Uq(sl2).