McKay Correspondence for semisimple Hopf actions on regular graded algebras, I

TL;DR

This paper extends the McKay correspondence to noncommutative regular algebras under semisimple Hopf actions, proving Auslander's theorem and relating fixed rings to Kleinian singularities.

Contribution

It generalizes Auslander's theorem for semisimple Hopf actions on noncommutative Artin-Schelter regular algebras of dimension two, linking fixed rings to Kleinian singularities.

Findings

Proves Auslander's theorem in a noncommutative setting.

Shows fixed rings correspond to analogues of Kleinian singularities.

Establishes a generalized McKay correspondence for Hopf actions.

Abstract

In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin-Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring A^H under such an action arises an analogue of a coordinate ring of a Kleinian singularity.