Symplectic wheelgebras and noncommutative geometry

TL;DR

This paper develops a framework for noncommutative differential geometry using symplectic wheelgebras, introducing generalized wheelspaces, and connecting them with classical structures via functors and a wheeled Cartan calculus.

Contribution

It introduces generalized wheelspaces forming a symmetric monoidal category, develops symplectic wheelgebras, and extends the Van den Bergh functor to bridge noncommutative and classical symplectic geometry.

Findings

Generalized wheelspaces form a symmetric monoidal category.

Symplectic wheelgebras are obtained from smooth bisymplectic algebras.

The wheeled Van den Bergh functor induces symplectic structures on representation schemes.

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.