Pointed Hopf actions on quantum generalized Weyl algebras

TL;DR

This paper classifies certain Hopf algebra actions on quantum Weyl algebras and shows their invariant rings are often commutative Kleinian singularities, advancing understanding of symmetries in noncommutative algebra.

Contribution

It provides a classification of inner-faithful pointed Hopf algebra actions on quantum generalized Weyl algebras respecting grading, and describes the nature of their invariant rings.

Findings

Classified inner-faithful actions of generalized Taft algebras on quantum Weyl algebras.

Proved that invariant rings are generically commutative Kleinian singularities.

Established conditions for actions to respect the grading structure.

Abstract

We study actions of pointed Hopf algebras in the $\ZZ$-graded setting. Our main result classifies inner-faithful actions of generalized Taft algebras on quantum generalized Weyl algebras which respect the $\ZZ$-grading. We also show that generically the invariant rings of Taft actions on quantum generalized Weyl algebras are commutative Kleinian singularities.