Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Shifted double Poisson structures and noncommutative Poisson extensions

Leilei Liu, Jieheng Zeng, Hu Zhao

TL;DR

The paper develops a comprehensive framework for shifted noncommutative Poisson geometry via double Poisson structures and their extensions, establishing that $n$-shifted double Poisson brackets on augmented dg algebras induce graded Lie structures on reduced cyclic homology and descend compatibly through noncommutative Hamiltonian reduction. It introduces the $\mathfrak{s}$-construction and a cobar–bar canonical resolution to transport double Poisson brackets to derived settings, proving independence of cofibrant resolutions and yielding a derived, homotopical perspective on NC Poisson structures. The Kontsevich–Rosenberg principle is extended to shifted, derived representation schemes, linking noncommutative data to shifted Poisson structures on derived moduli stacks of representations and their Cartan models. The work applies these developments to preprojective algebras, proving a canonical graded Lie algebra structure on the reduced cyclic homology of $\Pi Q$ and extending this structure along the cobar–bar construction to various deformations, with broader implications for quiver varieties and representation theory.

Abstract

We develop a theory of noncommutative Poisson extensions. For an augmented dg algebra \(A\), we show that any shifted double Poisson bracket on \(A\) induces a graded Lie algebra structure on the reduced cyclic homology. Under the Kontsevich--Rosenberg principle, we further prove that the noncommutative Poisson extension is compatible with noncommutative Hamiltonian reduction. Moreover, we show that shifted double Poisson structures are independent of the choice of cofibrant resolutions and that they induce shifted Poisson structures on the derived moduli stack of representations.

Shifted double Poisson structures and noncommutative Poisson extensions

TL;DR

The paper develops a comprehensive framework for shifted noncommutative Poisson geometry via double Poisson structures and their extensions, establishing that -shifted double Poisson brackets on augmented dg algebras induce graded Lie structures on reduced cyclic homology and descend compatibly through noncommutative Hamiltonian reduction. It introduces the -construction and a cobar–bar canonical resolution to transport double Poisson brackets to derived settings, proving independence of cofibrant resolutions and yielding a derived, homotopical perspective on NC Poisson structures. The Kontsevich–Rosenberg principle is extended to shifted, derived representation schemes, linking noncommutative data to shifted Poisson structures on derived moduli stacks of representations and their Cartan models. The work applies these developments to preprojective algebras, proving a canonical graded Lie algebra structure on the reduced cyclic homology of and extending this structure along the cobar–bar construction to various deformations, with broader implications for quiver varieties and representation theory.

Abstract

We develop a theory of noncommutative Poisson extensions. For an augmented dg algebra , we show that any shifted double Poisson bracket on induces a graded Lie algebra structure on the reduced cyclic homology. Under the Kontsevich--Rosenberg principle, we further prove that the noncommutative Poisson extension is compatible with noncommutative Hamiltonian reduction. Moreover, we show that shifted double Poisson structures are independent of the choice of cofibrant resolutions and that they induce shifted Poisson structures on the derived moduli stack of representations.

Paper Structure

This paper contains 17 sections, 36 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conventions and notation
  3. Noncommutative Poisson Geometry
  4. Double Poisson brackets
  5. Higher noncommutative Poisson structures
  6. Kontsevich--Rosenberg Principle
  7. Derived representation schemes
  8. Explicit presentation
  9. Higher traces
  10. Relation with stacks of representations
  11. Noncommutative Poisson Extensions
  12. Double Poisson algebra extension
  13. NC Poisson algebra extension
  14. Reduction commutes with extension
  15. $\mathfrak{s}$-constructions
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $Q$ be a finite quiver. There exists a dg Loday algebra structure on $\mathbf{\Omega}\mathbf{B}\Pi Q$. Furthermore, this dg Loday structure induces a graded Lie algebra structure on $\overline{\mathsf{HC}}_\bullet(\Pi Q)$.

Figures (1)

  • Figure 1: Quiver $Q$ and the doubled quiver $\overline{Q}$.

Theorems & Definitions (79)

  • Theorem 1.2: Theorem \ref{['thm: preproj']}
  • Theorem 1.3: Theorem \ref{['thm: com cubic']}
  • Theorem 1.4: Theorem \ref{['thm: cob bar has dpoiss']}, Corollary \ref{['cor: dpois to lie']}
  • Theorem 1.5: Theorem \ref{['thm: dpoiss not reso']}
  • Definition 2.1: Van den Bergh
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 69 more