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Poisson structures in the Banach setting: comparison of different approaches

Tomasz Goliński, Praful Rahangdale, Alice Barbora Tumpach

TL;DR

The paper surveys how Poisson geometry can be formulated on Banach manifolds via three main routes: global brackets on $\mathscr{C}^ ty(M)$, subalgebra-based brackets, and tensor-based constructions via subbundles of $T^*M$. It analyzes locality through localizability, the role of Hamiltonian vector fields, and the conditions under which a Poisson tensor $\sharp$ or a Schouten bracket $[\![\pi,\pi]\!]$ exists. Through concrete examples and a synthesis, it clarifies the trade-offs between generality and structural guarantees across frameworks, suggesting that a unifying co-distribution viewpoint may subsume existing definitions under appropriate regularity. The results illuminate how infinite-dimensional phenomena such as queer brackets challenge classical Poisson intuitions and guide the toolkit for infinite-dimensional Hamiltonian dynamics and integrable systems.

Abstract

In this paper we examine various approaches to the notion of Poisson manifold in the context of Banach manifolds. Existing definitions are presented and differences between them are explored and illustrated with examples.

Poisson structures in the Banach setting: comparison of different approaches

TL;DR

The paper surveys how Poisson geometry can be formulated on Banach manifolds via three main routes: global brackets on , subalgebra-based brackets, and tensor-based constructions via subbundles of . It analyzes locality through localizability, the role of Hamiltonian vector fields, and the conditions under which a Poisson tensor or a Schouten bracket exists. Through concrete examples and a synthesis, it clarifies the trade-offs between generality and structural guarantees across frameworks, suggesting that a unifying co-distribution viewpoint may subsume existing definitions under appropriate regularity. The results illuminate how infinite-dimensional phenomena such as queer brackets challenge classical Poisson intuitions and guide the toolkit for infinite-dimensional Hamiltonian dynamics and integrable systems.

Abstract

In this paper we examine various approaches to the notion of Poisson manifold in the context of Banach manifolds. Existing definitions are presented and differences between them are explored and illustrated with examples.

Paper Structure

This paper contains 12 sections, 3 theorems, 33 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Poisson structures defined for the whole algebra of smooth functions on a Banach manifold
  4. Poisson structures defined for subalgebras of smooth functions on a Banach manifold
  5. Poisson structures defined using bundle morphisms $P : T^\flat M \rightarrow TM$
  6. Poisson structures defined using Poisson algebras
  7. Poisson structures defined using Poisson tensors
  8. Comparison of different approaches to Poisson Geometry in infinite-dimensions
  9. Poisson algebra of functions vs subbundles of the cotangent bundle
  10. Localizability
  11. Examples of Poisson brackets
  12. Summary of the differences between previous approaches

Key Result

Proposition 8

The existence of bump functions on the modeling spaces of the manifold $M$ implies localizability of Poisson brackets (or more generally of derivations) on $M$.

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: Odzijewicz--Ratiu OR, Definition 2.1
  • Remark 5
  • Definition 6
  • Definition 7
  • Proposition 8
  • proof
  • Definition 9: Beltiţă--Goliński--Tumpach BGT
  • ...and 24 more