A Derived McKay Correspondence for Quantum Kleinian Singularities and Calabi-Yau Algebras of Global Dimension Three

The classical McKay correspondence establishes a profound connection between the representation theory of a finite subgroup of special matrices and the geometry of the minimal resolution of the corresponding Kleinian singularity. This paper extends this correspondence to the noncommutative realm of quantum algebraic geometry. We investigate quantum Kleinian singularities, defined as fixed rings of a quantum plane under the action of a finite cyclic group of automorphisms. These quantum planes are quadratic Artin-Schelter regular algebras of global dimension two and have a highly noncommutative structure, generally not admitting commutative crepant resolutions.