For $n\geq 4$ we shall construct a family $D(q)$ of non-commutative deformations of the coordinate algebra of a Kleinian singularity of type $D_n$ depending on a polynomial $q$ of degree $n$. We shall prove that every deformation of a type $D$ Kleinian singularity which is not commutative is isomorphic to some $D(q)$. We shall then consider in type $D$ the family of deformations $\mathcal{ O}^{ \boldsymbol{ \lambda}}$ constructed by Crawley-Boevey and Holland. For each $\mathcal{ O}^{ \boldsymbol{ \lambda}}$ which is not commutative we shall exhibit an explicit isomorphism $D(q)\cong \mathcal{ O}^{ \boldsymbol{ \lambda}}$ for a suitable choice of $q$. This will enable us to prove that every deformation of a Kleinian singularity of type $D_n$ is isomorphic to some $\mathcal{ O}^{ \boldsymbol{ \lambda}}$ and determine when two $\mathcal{ O}^{ \boldsymbol{ \lambda}}$ are isomorphic.
Categories: math.RA