Queer Poisson brackets

TL;DR

This paper introduces a method to construct Poisson brackets on Banach manifolds that depend on higher-order derivatives, challenging previous assumptions about the Leibniz property implying the existence of a Poisson tensor.

Contribution

It provides explicit examples of Poisson brackets on Banach manifolds that depend on higher derivatives, countering prior claims about the Leibniz property and Poisson tensors.

Findings

Constructed Poisson brackets depending on higher derivatives

Provided counterexamples to Leibniz property implying Poisson tensor

Discussed implications for Poisson geometry on Banach manifolds

Abstract

We give a method to construct Poisson brackets $\{\cdot,\cdot\}$ on Banach manifolds~$M$, for which the value of $\{f,g\}$ at some point $m\in M$ may depend on higher order derivatives of the smooth functions $f,g\colon M\to{\mathbb R}$, and not only on the first-order derivatives, as it is the case on all finite-dimensional manifolds. We discuss specific examples in this connection, as well as the impact on the earlier research on Poisson geometry of Banach manifolds. Those brackets are counterexamples to the claim that the Leibniz property for any Poisson bracket on a Banach manifold would imply the existence of a Poisson tensor for that bracket.