Pointed Hopf actions on fields, II

TL;DR

This paper classifies finite-dimensional pointed Galois-theoretical Hopf algebras of finite Cartan type, identifying conditions under which they act inner faithfully on fields, including certain quantum groups and their twists.

Contribution

It provides the first classification of Galois-theoretical Hopf algebras of finite Cartan type, detailing conditions for specific types and quantum group twists.

Findings

Classified pointed Galois-theoretical Hopf algebras of type A_1^{ imes r}

Identified conditions for rank two Hopf algebras to be Galois-theoretical

Established criteria for Reshetikhin twists of small quantum groups to possess this property

Abstract

This is a continuation of the authors' study of finite-dimensional pointed Hopf algebras H which act inner faithfully on commutative domains. As mentioned in Part I of this work, the study boils down to the case where H acts inner faithfully on a field. These Hopf algebras are referred to as Galois-theoretical. In this work, we provide classification results for finite-dimensional pointed Galois-theoretical Hopf algebras H of finite Cartan type. Namely, we determine when such H of type A_1^{\times r} and some H of rank two possess the Galois-theoretical property. Moreover, we provide necessary and sufficient conditions for Reshetikhin twists of small quantum groups to be Galois-theoretical.