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Isomorphisms of quantizations via quantization of resolutions

Ivan Losev

TL;DR

The work tackles the problem of relating quantizations of graded Poisson algebras arising from symplectic resolutions by constructing and comparing quantizations from three frameworks: symplectic reflection algebras, quantum Hamiltonian reductions, and W-algebras. It develops a unified approach based on deformations of symplectic resolutions and the noncommutative period map, enabling the authors to prove isomorphisms between diverse quantizations, including new W-algebra results and new proofs for SRA–QHR correspondences. Key contributions include the Klein_W and type_A theorems connecting parabolic W-algebras with quantum Hamiltonian reductions for quivers, and a detailed SRA_iso showing an explicit isomorphism between spherical SRAs for wreath products and quiver-variety reductions, via Procesi bundles and completions. The findings have representation-theoretic significance, linking geometric quantizations to algebraic constructions and offering a robust framework for translating between different quantization viewpoints in types Kleinian and A.

Abstract

In this paper we prove the existence of isomorphisms between certain non-commutative algebras that are interesting from representation theoretic perspective and arise as quantizations of certain Poisson algebras. We show that quantizations of Kleinian resolutions obtained by three different constructions are isomorphic to each other. The constructions are via symplectic reflection algebras, quantum Hamiltonian reduction, and W-algebras. Next, we prove that parabolic W-algebras in type A are isomorphic to quantum Hamiltonian reductions associated to quivers of type A. Finally, we show that the symplectic reflection algebras for wreath-products of the symmetric group and a Kleinian group are isomorphic to certain quantum Hamiltonian reductions. Our results involving W-algebras are new, while for those dealing with symplectic reflection algebras we just give new proofs. A key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate Poisson varieties.

Isomorphisms of quantizations via quantization of resolutions

TL;DR

The work tackles the problem of relating quantizations of graded Poisson algebras arising from symplectic resolutions by constructing and comparing quantizations from three frameworks: symplectic reflection algebras, quantum Hamiltonian reductions, and W-algebras. It develops a unified approach based on deformations of symplectic resolutions and the noncommutative period map, enabling the authors to prove isomorphisms between diverse quantizations, including new W-algebra results and new proofs for SRA–QHR correspondences. Key contributions include the Klein_W and type_A theorems connecting parabolic W-algebras with quantum Hamiltonian reductions for quivers, and a detailed SRA_iso showing an explicit isomorphism between spherical SRAs for wreath products and quiver-variety reductions, via Procesi bundles and completions. The findings have representation-theoretic significance, linking geometric quantizations to algebraic constructions and offering a robust framework for translating between different quantization viewpoints in types Kleinian and A.

Abstract

In this paper we prove the existence of isomorphisms between certain non-commutative algebras that are interesting from representation theoretic perspective and arise as quantizations of certain Poisson algebras. We show that quantizations of Kleinian resolutions obtained by three different constructions are isomorphic to each other. The constructions are via symplectic reflection algebras, quantum Hamiltonian reduction, and W-algebras. Next, we prove that parabolic W-algebras in type A are isomorphic to quantum Hamiltonian reductions associated to quivers of type A. Finally, we show that the symplectic reflection algebras for wreath-products of the symmetric group and a Kleinian group are isomorphic to certain quantum Hamiltonian reductions. Our results involving W-algebras are new, while for those dealing with symplectic reflection algebras we just give new proofs. A key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate Poisson varieties.

Paper Structure

This paper contains 33 sections, 43 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Content of the paper
  3. Conventions and notation
  4. Deformations of symplectic schemes
  5. Commutative deformations
  6. Generalities on deformation quantization
  7. Non-commutative period map
  8. Hamiltonian reduction
  9. Classical reduction
  10. The Duistermaat-Heckman theorem
  11. Quantum reduction: algebra level
  12. Quantum reduction: sheaf level
  13. Symplectic resolutions
  14. Generalities
  15. Quiver varieties
  16. ...and 18 more sections

Key Result

Proposition 2.1.1

Recall that $X_0$ is a symplectic variety with $H^1(X_0,\mathcal{O}_{X_0})=H^2(X_0,\mathcal{O}_{X_0})=\{0\}$. The map $X/S\mapsto p_{X/S}$ is a bijection between The inverse map assigns to $p:S\rightarrow \operatorname{Per}$ the pull-back $p^*(X_{univ}/\operatorname{Per})$ of some universal deformation $X_{univ}/\operatorname{Per}$.

Theorems & Definitions (78)

  • Proposition 2.1.1
  • Proposition 2.1.2
  • Proposition 2.3.1
  • Proposition 2.3.2
  • Corollary 2.3.3
  • Lemma 2.3.4
  • Proposition 2.3.5: BK1, Proposition 2.7
  • Definition 2.3.6
  • proof : Proof of Proposition \ref{['Prop:Per_group']}
  • Remark 2.3.7
  • ...and 68 more