Isomorphism Problems of Noncommutative Deformations of Type D Kleinian Singularities

Abstract

We construct all possible noncommutative deformations of a Kleinian singularity ${\mathbb C}^2/Γ$ 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 $N_{\SL(2)}(Γ)$ on ${\mathbb C}/Γ$. We deduce that the moduli space of isomorphism classes of noncommutative deformations in type $D_n$ is isomorphic to a vector space of dimension $n$.