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Partial Poisson Lie groups and groupoids. Application to Von Neumann algebras

Fernand Pelletier, Patrick Cabau

TL;DR

The paper develops a comprehensive framework for partial Poisson and sub-Poisson geometry in the convenient (often infinite-dimensional) setting, extending Weinstein’s finite-dimensional Poisson-Lie theory to Banach and broader convenient spaces. It systematically builds the theory from partial Poisson manifolds through to convenient Lie groups and groupoids, introducing linear anchors $P$ and bivector data $oldsymbol\Lambda$ that satisfy Jacobi-type conditions via Schouten brackets, and showing how these structures induce partial bialgebroids and dual pairs. A central theme is the use of direct/indirect limits to construct new examples of (partial) Poisson-Lie groups and groupoids that are not Banach-Lie in the classical sense, while maintaining a coherent infinitesimal-to-global correspondence via algebroid duals and conormal bundles. The work further connects this framework to operator algebras, illustrating how Odzijewicz-type groupoids associated to Von Neumann algebras fit into the partial Poisson-Convenient setting, and it lays groundwork for potential applications in infinite-dimensional Poisson geometry and quantum groups. Overall, the manuscript provides a self-contained, rigorous extension of finite-dimensional Poisson-Lie theory to convenient/ Banach contexts, including detailed constructions of multiplicative forms, coisotropic/lagrangian relations, and direct-limit groupoid examples with implications for Von Neumann algebra theory.

Abstract

The purpose of this paper is to propose a version of the notion of convenient Lie groupoid as a generalization of this concept in finite dimension. The authors point out which obstructions appear in the infinite dimensional context and how an adapted notion of "bi-algebroid " in finite dimension (cf. \cite{MaXu94}) can be, nevertheless, recovered. This paper is self-contained and recalls some important properties of "partial Poisson manifolds" (cf. \cite{CaPe23}, Chapter~7) and "Banach Poisson Lie groups" (cf. \cite{Tum20}) needed for their purpose. The paper also gives an illustration of these concepts from all the results of A.~Odzijewicz and his collaborators on Lie groupoids and Von Neumann algebras.

Partial Poisson Lie groups and groupoids. Application to Von Neumann algebras

TL;DR

The paper develops a comprehensive framework for partial Poisson and sub-Poisson geometry in the convenient (often infinite-dimensional) setting, extending Weinstein’s finite-dimensional Poisson-Lie theory to Banach and broader convenient spaces. It systematically builds the theory from partial Poisson manifolds through to convenient Lie groups and groupoids, introducing linear anchors and bivector data that satisfy Jacobi-type conditions via Schouten brackets, and showing how these structures induce partial bialgebroids and dual pairs. A central theme is the use of direct/indirect limits to construct new examples of (partial) Poisson-Lie groups and groupoids that are not Banach-Lie in the classical sense, while maintaining a coherent infinitesimal-to-global correspondence via algebroid duals and conormal bundles. The work further connects this framework to operator algebras, illustrating how Odzijewicz-type groupoids associated to Von Neumann algebras fit into the partial Poisson-Convenient setting, and it lays groundwork for potential applications in infinite-dimensional Poisson geometry and quantum groups. Overall, the manuscript provides a self-contained, rigorous extension of finite-dimensional Poisson-Lie theory to convenient/ Banach contexts, including detailed constructions of multiplicative forms, coisotropic/lagrangian relations, and direct-limit groupoid examples with implications for Von Neumann algebra theory.

Abstract

The purpose of this paper is to propose a version of the notion of convenient Lie groupoid as a generalization of this concept in finite dimension. The authors point out which obstructions appear in the infinite dimensional context and how an adapted notion of "bi-algebroid " in finite dimension (cf. \cite{MaXu94}) can be, nevertheless, recovered. This paper is self-contained and recalls some important properties of "partial Poisson manifolds" (cf. \cite{CaPe23}, Chapter~7) and "Banach Poisson Lie groups" (cf. \cite{Tum20}) needed for their purpose. The paper also gives an illustration of these concepts from all the results of A.~Odzijewicz and his collaborators on Lie groupoids and Von Neumann algebras.
Paper Structure (68 sections, 61 theorems, 202 equations)

This paper contains 68 sections, 61 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Some Essential Properties of Partial Poisson Manifolds
  3. Relations in Linear Poisson and Symplectic Convenient Spaces
  4. Partial Linear Poisson Convenient Spaces
  5. Linear Poisson (symplectic) relations
  6. Basic Properties of Partial Poisson Manifolds
  7. Partial Poisson Manifolds
  8. Poisson bracket associated to a convenient Lie algebroid
  9. Characteristic distribution and foliation
  10. Partial Lie Algebroids and Partial Poisson Manifolds
  11. Jacobi Identity and Schouten Condition
  12. Partial Poisson Bialgebroid and its Associated Cohomology
  13. Partial Bialgebroid
  14. Convenient Linear Partial Poisson Spaces
  15. Poisson morphisms and Poisson maps
  16. ...and 53 more sections

Key Result

Lemma 2.5

${}$

Theorems & Definitions (138)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Proposition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Definition 2.10
  • Proposition 2.11
  • ...and 128 more