Isomorphism Problems of Noncommutative Deformations of Type D Kleinian Singularities

TL;DR

This paper classifies all noncommutative deformations of type D Kleinian singularities, showing that their isomorphism classes form a vector space of dimension n, and characterizing when two deformations are equivalent.

Contribution

It constructs all noncommutative deformations of type D Kleinian singularities and determines the conditions for their isomorphisms, linking them to group actions.

Findings

All deformations are classified by a vector space of dimension n.

Isomorphisms correspond to the action of the normalizer group.

The moduli space of deformations is explicitly described.

Abstract

We construct all possible noncommutative deformations of a Kleinian singularity ${\mathbb C}^2/\Gamma$ of type $D_n$ in terms of generators and relations, and solve the problem of when two deformations are isomorphic. We prove that all isomorphisms arise naturally from the action of the normalizer $N_{\SL(2)}(\Gamma)$ on ${\mathbb C}/\Gamma$. We deduce that the moduli space of isomorphism classes of noncommutative deformations in type $D_n$ is isomorphic to a vector space of dimension $n$.