Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Noncommutative Hamiltonian structures and quantizations on preprojective algebras

Hu Zhao

TL;DR

This work develops a noncommutative Hamiltonian framework built from double Poisson structures and a noncommutative moment map to study quantization and reduction in quiver settings. It proves that quantization commutes with reduction in the noncommutative sense by constructing a two-term complex whose first cohomology yields the quantized reduced algebra, and it introduces a semidirect product $A\rtimes\mathcal{G}^A$ linking equivariant sheaves to modules. In the quiver context, the authors relate noncommutative quantizations of the preprojective algebra $\Pi Q$ to quantizations of the corresponding quiver varieties via quantum trace maps, with an explicit deformation parameter $\mathbf{r}$ encoding higher-order information. The results extend to deformed preprojective algebras $\Pi^{\lambda} Q$ and provide a Morita-invariant, diagrammatic framework that aligns noncommutative and classical reductions under the Kontsevich–Rosenberg principle.

Abstract

Given a noncommutative Hamiltonian space $A$, we prove that the conjecture ``{\it quantization commutes with reduction}'' holds for $A$. We further construct a semidirect product algebra $A \rtimes \mG^A$, and establish a correspondence between equivariant sheaves on the representation space and left $A\rtimes\mG^A$-modules. In the quiver setting, using the quantum and classical trace maps, we establish the explicit correspondence between quantizations of a preprojective algebra and those of a quiver variety.

Noncommutative Hamiltonian structures and quantizations on preprojective algebras

TL;DR

This work develops a noncommutative Hamiltonian framework built from double Poisson structures and a noncommutative moment map to study quantization and reduction in quiver settings. It proves that quantization commutes with reduction in the noncommutative sense by constructing a two-term complex whose first cohomology yields the quantized reduced algebra, and it introduces a semidirect product linking equivariant sheaves to modules. In the quiver context, the authors relate noncommutative quantizations of the preprojective algebra to quantizations of the corresponding quiver varieties via quantum trace maps, with an explicit deformation parameter encoding higher-order information. The results extend to deformed preprojective algebras and provide a Morita-invariant, diagrammatic framework that aligns noncommutative and classical reductions under the Kontsevich–Rosenberg principle.

Abstract

Given a noncommutative Hamiltonian space , we prove that the conjecture ``{\it quantization commutes with reduction}'' holds for . We further construct a semidirect product algebra , and establish a correspondence between equivariant sheaves on the representation space and left -modules. In the quiver setting, using the quantum and classical trace maps, we establish the explicit correspondence between quantizations of a preprojective algebra and those of a quiver variety.
Paper Structure (12 sections, 24 theorems, 89 equations, 3 figures)

This paper contains 12 sections, 24 theorems, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Noncommutative Hamiltonian spaces
  3. Noncommutative Hamiltonian spaces
  4. Equivariant sheaves and the semidirect product algebra
  5. Noncommutative quantum reduction
  6. Quantization of preprojective algebras
  7. Noncommutative Hamiltonian structure on the quiver algebra
  8. Schedler's quantization
  9. Noncommutative quantum reduction in the quiver setting
  10. [Q, R]=0 on quiver representation spaces
  11. Quantum trace maps
  12. The deformed case

Key Result

Theorem 1.1

Let $(A,\mathopen{\{\!\!\{}-,-\mathclose{\}\!\!\}},\mathbf{w})$ be a noncommutative Hamiltonian space. For any $N\in\mathbb{N}$ and $\mathrm{GL}_N(\mathbb{K})$-equivariant $\mathcal{O}$-module $\mathcal{F}$, the sheaf $\mathcal{E}_N\otimes_{\mathcal{O}}\mathcal{F}$ naturally carries a left $A\rtimes

Figures (3)

  • Figure 1: Quiver $Q$ and its doubled version $\overline{Q}$.
  • Figure :
  • Figure :

Theorems & Definitions (55)

  • Theorem 1.1: Theorem \ref{['thm: equ.vdb']}
  • Theorem 1.2: Theorem \ref{['thm: quantization by a complex']}
  • Theorem 1.3: Theorem \ref{['thm: noncom quant moment map']}
  • Lemma 1.4: Lemma \ref{['lem: q.trace preserves ideals']}
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6: Van den Bergh
  • ...and 45 more