Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Compatible Poisson structures on multiplicative quiver varieties

Maxime Fairon

TL;DR

The paper constructs a Hamiltonian quasi-Poisson pencil on open representation spaces of a doubled quiver by combining Van den Bergh’s quasi-Poisson structure with commuting C^×-actions, producing a family of compatible Poisson structures that descend to multiplicative quiver varieties under reduction. It develops a general framework for quasi-Poisson pencils, including abelian-action and center-embedding tricks, and applies this to various settings such as character varieties and (additive) quiver varieties, providing explicit formulas for the corresponding bivectors and 2-forms. A key noncommutative origin is given via double Poisson theory, yielding a universal Poisson structure that induces the pencil on MQVs. The framework is then applied to the spin Ruijsenaars–Schneider phase space, establishing a master quasi-Poisson space whose reduction yields a Poisson pencil containing the CF and AO structures, furnishing local parametrizations, Hamiltonian formulations, and integrability results in this rich setting.

Abstract

Any multiplicative quiver variety is endowed with a Poisson structure constructed by Van den Bergh through reduction from a Hamiltonian quasi-Poisson structure. The smooth locus carries a corresponding symplectic form defined by Yamakawa through quasi-Hamiltonian reduction. In this note, we include the Poisson structure as part of a pencil of compatible Poisson structures on the multiplicative quiver variety. The pencil is defined by reduction from a pencil of Hamiltonian quasi-Poisson structures which has dimension $\ell(\ell-1)/2$, where $\ell$ is the number of arrows in the underlying quiver. For each element of the pencil, we exhibit the corresponding compatible symplectic or quasi-Hamiltonian structure. We comment on analogous constructions for character varieties and quiver varieties. This formalism is applied to the spin Ruijsenaars-Schneider phase space in order to explain the compatibility of two Poisson structures that have recently appeared in the literature.

Compatible Poisson structures on multiplicative quiver varieties

TL;DR

The paper constructs a Hamiltonian quasi-Poisson pencil on open representation spaces of a doubled quiver by combining Van den Bergh’s quasi-Poisson structure with commuting C^×-actions, producing a family of compatible Poisson structures that descend to multiplicative quiver varieties under reduction. It develops a general framework for quasi-Poisson pencils, including abelian-action and center-embedding tricks, and applies this to various settings such as character varieties and (additive) quiver varieties, providing explicit formulas for the corresponding bivectors and 2-forms. A key noncommutative origin is given via double Poisson theory, yielding a universal Poisson structure that induces the pencil on MQVs. The framework is then applied to the spin Ruijsenaars–Schneider phase space, establishing a master quasi-Poisson space whose reduction yields a Poisson pencil containing the CF and AO structures, furnishing local parametrizations, Hamiltonian formulations, and integrability results in this rich setting.

Abstract

Any multiplicative quiver variety is endowed with a Poisson structure constructed by Van den Bergh through reduction from a Hamiltonian quasi-Poisson structure. The smooth locus carries a corresponding symplectic form defined by Yamakawa through quasi-Hamiltonian reduction. In this note, we include the Poisson structure as part of a pencil of compatible Poisson structures on the multiplicative quiver variety. The pencil is defined by reduction from a pencil of Hamiltonian quasi-Poisson structures which has dimension , where is the number of arrows in the underlying quiver. For each element of the pencil, we exhibit the corresponding compatible symplectic or quasi-Hamiltonian structure. We comment on analogous constructions for character varieties and quiver varieties. This formalism is applied to the spin Ruijsenaars-Schneider phase space in order to explain the compatibility of two Poisson structures that have recently appeared in the literature.
Paper Structure (31 sections, 33 theorems, 122 equations)

This paper contains 31 sections, 33 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Original Poisson structure on multiplicative quiver varieties
  3. Notation
  4. Quasi-Poisson and quasi-Hamiltonian geometry
  5. Basics
  6. Definitions and standard results
  7. Multiplicative quiver variety
  8. Quiver representations
  9. Definition
  10. (Quasi-)Poisson structure and corresponding 2-form
  11. The (quasi-)Poisson pencil and its variants
  12. General framework
  13. First properties
  14. Construction using an abelian group action
  15. The trick: embeddings of $\mathbb{C}^\times$ in the center of $G$
  16. ...and 16 more sections

Key Result

Theorem A

Set $r_Q:=\frac{1}{2}|Q|(|Q|-1)$. Then the $\operatorname{GL}_{\mathbf{n}}$-variety $M_{\overline{Q}}^\circ$ admits a Hamiltonian quasi-Poisson pencil of order $r \leq r_Q$ centered at the quasi-Poisson bivector $P$Eq:qP-MQV constructed by Van den Bergh VdB1, with moment map $\Phi$Eq:Intro1. If $n_s

Theorems & Definitions (69)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Proposition 2.6: Fusion
  • Definition 2.7: CBShawYa
  • Theorem 2.8: VdB1VdB2Ya
  • ...and 59 more