Analytic Poisson brackets on rational functions on the Riemann sphere and their generalization

TL;DR

This paper introduces a hierarchy of Poisson structures on rational functions on the Riemann sphere, providing a direct proof of the Jacobi identity and exploring related hierarchies and Darboux coordinates.

Contribution

It offers a direct proof of the Jacobi identity for these Poisson brackets, advancing understanding of their structure and generalizations.

Findings

Direct proof of Jacobi identity for the hierarchy

Construction of Darboux coordinates for a related hierarchy

Example of a new hierarchy of Poisson brackets

Abstract

We consider a hierarchy of Poisson structures defined on rational functions on the Riemann sphere. This hierarchy is originated in the theory of the integrable Camassa-Holm equation associated with the Krein's string spectral problem. Previously the proof of Jacobi identity was obtained by reducing the bracket to canonical Darboux coordinates. The main result of this note is a direct proof of the Jacobi identity. It turns out that the direct proof of the Jacobi identity is far from trivial. We also give an example of another hierarchy of Poisson brackets and construct Darboux coordinates for it.