# Algebra Geometry Seminar: Modules over Noncommutative Deformations of Kleinian Singularities

**Author: Lona**

**Author: Lona**

Speaker: Jonas Hartwig (ISU)

Abstract:

A *Kleinian* (or *du Val*, or *simple surface*) *singularity* XX is the set of orbits C2/GC2/G where GG is a finite group of 2×2 matrices of determinant 11. The set XX is a two-dimensional affine variety whose ring of regular functions is C[X]=C[x,y]GC[X]=C[x,y]G, the ring of all GG–invariant polynomials in two variables. Since the action of GGon C2C2 is not free (origin is fixed), XX is a singular variety, hence the name. Kleinian singularities follow an ADE classification pattern.

Kleinian singularities lie at the crossroads of an overwhelming number of topics in geometry, algebra, and representation theory. As an example, the celebrated McKay correspondence gives a baffling connection between the sequence of blow-ups when resolving the singular point in XX, and tensor products of irreducible representations of GG.

In this talk we will focus on the *deformation quantization* OλOλ of C[X]C[X] proposed by Crawley-Boevey and Holland in 1998. In the Type D case, for which Boddington provided an embedding OλOλ into a skew-group algebra. By adjusting this embedding, we obtain that OλOλ are examples of so called *principal Galois orders*, a large class of algebras that include the enveloping algebras of glngln. This allows us to classify all irreducible Harish-Chandra modules over OλOλ. Furthermore we obtain a possibly new realization for OλOλ in Type D, in terms of Bernstein-Gelfand-Gelfand divided difference operators.