Automorphism groups of deformations and quantizations of Kleinian singularities

TL;DR

This paper explores the automorphism groups of deformations and quantizations of Kleinian singularities of types A and D, revealing their structures and the correspondence between isomorphisms of these mathematical objects.

Contribution

It establishes the equivalence of automorphisms of deformations and quantizations for types A and D, and describes their automorphism groups' structures, including new computations for type D.

Findings

Automorphisms of deformations and quantizations are essentially the same for types A and D.

Automorphism groups have specific algebraic structures: amalgamated free product for type A, subgroups of Dynkin diagram automorphisms for type D.

All affine isomorphisms between deformations of type D are classified.

Abstract

It is known that, for the algebra of functions on a Kleinian singularity, the parameter space of deformations and the parameter space of quantizations coincide. We prove that, for a Kleinian singularity of type $\mathbf{A}$ or $\mathbf{D}$, isomorphisms between the quantizations are essentially the same as Poisson isomorphisms between deformations. In particular, the group of automorphisms of the deformation and the quantization corresponding to the same deformation parameter are isomorphic. We additionally describe the groups of automorphisms as abstract groups: for type $\mathbf{A}$ they have an amalgamated free product structure, for type $\mathbf{D}$ they are subgroups of the groups of Dynkin diagram automorphisms. For type $\mathbf{D}$ we also compute all the possible affine isomorphisms between deformations; this was not known before.