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Automorphism groups of deformations and quantizations of Kleinian singularities

Simone Castellan

TL;DR

The paper addresses how automorphisms of deformations and quantizations of Kleinian singularities relate, proving that for types ${\mathbf{A}}$ and ${\mathbf{D}}$ the isomorphisms between quantizations correspond to Poisson isomorphisms between deformations. It develops a two-pronged approach: in type ${\mathbf{A}}$ it establishes an amalgamated free product structure for automorphism groups and shows Iso and PIso groupoids coincide; in type ${\mathbf{D}}$ it uses a geometric compactification to classify all affine isomorphisms of deformations and proves Iso$({\mathcal{D}}_n)\cong\mathrm{PIso}({\mathcal{D}}_n)$, including an $S_3$-structure in the $n=4$ undeformed case. The results confirm that all (Poisson) isomorphisms arise from graded automorphisms of the base singularity, with only filtered or diagram-symmetry contributions in type ${\mathbf{D}}$, and no wild non-filtered isomorphisms appear. These findings connect to Namikawa/Losev’s work on deformation/quantization moduli and extend the Belov–Kontsevich paradigm to a broader class of symplectic singularities, clarifying the symmetry landscape for Kleinian cases.

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.

Automorphism groups of deformations and quantizations of Kleinian singularities

TL;DR

The paper addresses how automorphisms of deformations and quantizations of Kleinian singularities relate, proving that for types and the isomorphisms between quantizations correspond to Poisson isomorphisms between deformations. It develops a two-pronged approach: in type it establishes an amalgamated free product structure for automorphism groups and shows Iso and PIso groupoids coincide; in type it uses a geometric compactification to classify all affine isomorphisms of deformations and proves Iso, including an -structure in the undeformed case. The results confirm that all (Poisson) isomorphisms arise from graded automorphisms of the base singularity, with only filtered or diagram-symmetry contributions in type , and no wild non-filtered isomorphisms appear. These findings connect to Namikawa/Losev’s work on deformation/quantization moduli and extend the Belov–Kontsevich paradigm to a broader class of symplectic singularities, clarifying the symmetry landscape for Kleinian cases.

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 or , 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 they have an amalgamated free product structure, for type they are subgroups of the groups of Dynkin diagram automorphisms. For type we also compute all the possible affine isomorphisms between deformations; this was not known before.
Paper Structure (10 sections, 41 theorems, 139 equations)

This paper contains 10 sections, 41 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Type A
  4. Deformations and quantizations
  5. Amalgamated free product group structure
  6. The group of Poisson automorphisms of ${\mathcal{A}}(P)$
  7. Isomorphism groupoids (type $\mathbf{A}$)
  8. Type D
  9. Deformations and quantizations
  10. Isomorphisms of deformations

Key Result

Theorem A

Let $V/\Gamma$ be a Kleinian singularity of type $\mathbf{A}$ or $\mathbf{D}$. Then Conjecture conj main holds. In particular, the group of automorphisms of the quantization and of Poisson automorphisms of the deformation corresponding to the same parameter are isomorphic.

Theorems & Definitions (85)

  • Conjecture 1: belov-kanelAutomorphismsWeylAlgebra2005
  • Conjecture 2
  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 1
  • Definition 2
  • Remark 3
  • Proposition 4
  • proof
  • ...and 75 more