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Symplectic wheelgebras and noncommutative geometry

David Fernández, Estanislao Herscovich

TL;DR

The paper advances noncommutative differential geometry by replacing the non-monoidal wheelspace category with a symmetric monoidal framework of generalized wheelspaces, enabling a robust wheeled calculus. It develops the Fock wheelgebra and a Cartan calculus in the wheeled context, and proves that the wheeled Van den Bergh functor aligns with the classical KR principle, bridging noncommutative and commutative representation-theoretic structures. By establishing that smooth bisymplectic algebras yield symplectic Fock wheelgebras, it connects double Poisson and bisymplectic geometry to wheeled symplectic structures, with a concrete pathway to representation schemes. The results collectively provide a coherent, functorial route from noncommutative differential geometry to representation-theoretic algebro-geometric structures, and suggest broad applicability to deformation theory and quantization in a wheeled setting.

Abstract

In this article, we explore the following statement made by V. Ginzburg and T. Schedler in [Selecta Math. (N.S.) 16 (2010), no. 4, 673-730]: "an adequate framework for doing noncommutative differential geometry is provided by the notion of wheelspace. Wheelspaces form a symmetric monoidal category". However, the category of wheelspaces turns out not to be monoidal. To address this, we introduce generalized wheelspaces, which do form a symmetric monoidal category and provide solid ground for the theory of wheelgebras. To support their first claim, Ginzburg and Schedler defined Poisson (Fock) wheelgebras in connection with Van den Bergh's double Poisson algebras via the Fock functor. We provide strong evidence to their claim by introducing symplectic wheelgebras and prove that the Fock functor sends smooth bisymplectic algebras, as defined by W. Crawley-Boevey, V. Ginzburg and P. Etingof, into our symplectic wheelgebras. In the process, we develop a Cartan calculus adapted to this wheeled context. Moreover, we present a wheeled version of the significant Van den Bergh functor, which facilitates a formalization of the Kontsevich-Rosenberg principle, bridging the noncommutative and commutative frameworks. After establishing that the classical Van den Bergh functor factors through our wheeled version, we show that symplectic Fock wheelgebras naturally induce symplectic algebras on representation schemes.

Symplectic wheelgebras and noncommutative geometry

TL;DR

The paper advances noncommutative differential geometry by replacing the non-monoidal wheelspace category with a symmetric monoidal framework of generalized wheelspaces, enabling a robust wheeled calculus. It develops the Fock wheelgebra and a Cartan calculus in the wheeled context, and proves that the wheeled Van den Bergh functor aligns with the classical KR principle, bridging noncommutative and commutative representation-theoretic structures. By establishing that smooth bisymplectic algebras yield symplectic Fock wheelgebras, it connects double Poisson and bisymplectic geometry to wheeled symplectic structures, with a concrete pathway to representation schemes. The results collectively provide a coherent, functorial route from noncommutative differential geometry to representation-theoretic algebro-geometric structures, and suggest broad applicability to deformation theory and quantization in a wheeled setting.

Abstract

In this article, we explore the following statement made by V. Ginzburg and T. Schedler in [Selecta Math. (N.S.) 16 (2010), no. 4, 673-730]: "an adequate framework for doing noncommutative differential geometry is provided by the notion of wheelspace. Wheelspaces form a symmetric monoidal category". However, the category of wheelspaces turns out not to be monoidal. To address this, we introduce generalized wheelspaces, which do form a symmetric monoidal category and provide solid ground for the theory of wheelgebras. To support their first claim, Ginzburg and Schedler defined Poisson (Fock) wheelgebras in connection with Van den Bergh's double Poisson algebras via the Fock functor. We provide strong evidence to their claim by introducing symplectic wheelgebras and prove that the Fock functor sends smooth bisymplectic algebras, as defined by W. Crawley-Boevey, V. Ginzburg and P. Etingof, into our symplectic wheelgebras. In the process, we develop a Cartan calculus adapted to this wheeled context. Moreover, we present a wheeled version of the significant Van den Bergh functor, which facilitates a formalization of the Kontsevich-Rosenberg principle, bridging the noncommutative and commutative frameworks. After establishing that the classical Van den Bergh functor factors through our wheeled version, we show that symplectic Fock wheelgebras naturally induce symplectic algebras on representation schemes.

Paper Structure

This paper contains 45 sections, 39 theorems, 312 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and conventions
  4. Categorical definitions
  5. Algebras
  6. Modules and bimodules
  7. Rings over algebras
  8. Lie and Poisson algebras
  9. Restriction and induction
  10. Basic constructions for symmetric groups
  11. S-modules and diagonal S-bimodules
  12. Generalities
  13. Definitions of S-modules and diagonal S-bimodules, and basic properties
  14. Categorical properties: symmetric monoidal structures
  15. Shifted S-modules and diagonal S-bimodules
  16. ...and 30 more sections

Key Result

Lemma 3.7

The Cauchy tensor product $\otimes_{{\mathbb{S}}}$ given in eq:cauchy defines a monoidal product on the category ${\mathbb{S}}$-modules, whose unit is the unique ${\mathbb{S}}$-module $\mathbf{1}_{{\mathbb{S}}}$ such that $\mathbf{1}_{{\mathbb{S}}}(n) = 0$ for all $n \in {\mathbb{N}}$ and $\mathbf{1

Figures (1)

  • Figure 1: Left: Pictorial representation of a diagonal ${\mathbb{S}}$-bimodule. Right: Pictorial representation of a wheelspace with trace/contraction operation ${}^{\space n}_{n-1}t_2$.

Theorems & Definitions (140)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.5
  • proof
  • Remark 2.7
  • Example 2.8
  • Example 2.9
  • Example 2.10: Square-zero construction
  • Example 2.13
  • ...and 130 more