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Poisson structure on predual of Banach Lie algebroid

Tomasz Goliński, Grzegorz Jakimowicz

TL;DR

The paper addresses how to define a linear Poisson structure on the predual bundle $E_*$ of a Banach Lie algebroid and how this relates to Banach Poisson geometry. It develops a canonical, linear, localizable Poisson bracket on $E_*$ using two natural families of functions $f\circ\\pi_*$ and $\\lambda_X$, derives the Poisson tensor $\\Pi$, and provides explicit coordinate formulas that yield a Banach Poisson structure under suitable compatibility conditions. It also shows the reverse direction: from a linear Poisson bracket on $E_*$ one recovers a Banach Lie algebroid on $E$ via $[X_1,X_2] = \\lambda^{-1}(\\{\\lambda_{X_1},\\lambda_{X_2}\\})$ and $a(X) = T\\pi_* \\sharp(d\\lambda_X)$, establishing a Banach analogue of the Lie–Poisson correspondence. The paper includes illustrative examples, notably a weak symplectic structure on the precotangent bundle and a concrete trivial-bundle construction on $\\ell^2 \times \\ell^\infty$, clarifying the theory and highlighting conditions under which the predual carries a Banach Poisson structure. Together, these results extend the finite-dimensional Poisson–Lie algebroid correspondence to the Banach setting and offer practical criteria and models for infinite-dimensional Poisson geometry and potential applications in integrable systems.

Abstract

We construct the linear Poisson structure on the predual bundle of a Banach Lie algebroid. It is an alternative approach to the already known results on the linear sub-Poisson structure on the dual bundle. We also discuss the existence of queer Banach Lie algebroids. An example of a precotangent bundle is presented.

Poisson structure on predual of Banach Lie algebroid

TL;DR

The paper addresses how to define a linear Poisson structure on the predual bundle of a Banach Lie algebroid and how this relates to Banach Poisson geometry. It develops a canonical, linear, localizable Poisson bracket on using two natural families of functions and , derives the Poisson tensor , and provides explicit coordinate formulas that yield a Banach Poisson structure under suitable compatibility conditions. It also shows the reverse direction: from a linear Poisson bracket on one recovers a Banach Lie algebroid on via and , establishing a Banach analogue of the Lie–Poisson correspondence. The paper includes illustrative examples, notably a weak symplectic structure on the precotangent bundle and a concrete trivial-bundle construction on , clarifying the theory and highlighting conditions under which the predual carries a Banach Poisson structure. Together, these results extend the finite-dimensional Poisson–Lie algebroid correspondence to the Banach setting and offer practical criteria and models for infinite-dimensional Poisson geometry and potential applications in integrable systems.

Abstract

We construct the linear Poisson structure on the predual bundle of a Banach Lie algebroid. It is an alternative approach to the already known results on the linear sub-Poisson structure on the dual bundle. We also discuss the existence of queer Banach Lie algebroids. An example of a precotangent bundle is presented.
Paper Structure (11 sections, 13 theorems, 73 equations)

This paper contains 11 sections, 13 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Banach Poisson manifolds
  4. Banach Lie algebroids
  5. Correspondence between Poisson and Lie algebroid structures
  6. Local picture
  7. Poisson structure on predual to Lie algebroid
  8. Poisson tensor
  9. Banach Lie algebroid structure on dual of linear Poisson manifold
  10. Example: weak symplectic form on a precotangent bundle
  11. Example: trivial bundle $\ell^2 \times \ell^\infty\rightarrow \ell^2$

Key Result

Theorem 2.4

If the typical fiber $\mathbb E$ of the Banach vector bundle $E\rightarrow M$ admits a Schauder basis, then there are no queer Banach Lie algebroids on $E$.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • Remark 2.6
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • ...and 19 more