Noncommutative Deformations of Type-A Kleinian Singularities

Abstract

The type-A Kleinian singularities are the surfaces of the form C2/G where G is a finite cyclic subgroup of SL2(C). The standard noncommutative analog is the fixed ring of the Weyl algebra A1(C) under the induced action of G. These fixed rings are, however, only part of a large family of algebras whose associated graded ring is the coordinate ring of such a singularity. The structure of these algebras is studied with reference to the structure of the associated singularity and to the structure of primitive factor rings of U(sl(2, C)).