Hopf–Galois structures on ambiskew polynomial rings

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 generalized ambiskew ring, based on a suitable Hopf algebra $\underline{H}(R)$. Several examples are examined, including the Hopf–Galois objects over $U_q({\mathfrak{sl}}_2)$.