Inequalities from Poisson brackets

TL;DR

This paper introduces a tropicalization framework for Poisson structures, linking them to polyhedral cones and integrable systems, exemplified by the Gelfand-Zeitlin system on dual Poisson-Lie groups.

Contribution

It develops a novel tropicalization method for Poisson structures, connecting algebraic and geometric aspects, and demonstrates its application to classical integrable systems.

Findings

Tropicalization associates polyhedral cones to Poisson structures.

The integrable system on the dual Poisson-Lie group is isomorphic to Gelfand-Zeitlin system.

The framework applies to both real and complex Poisson manifolds.

Abstract

We introduce the notion of tropicalization for Poisson structures on $\mathbb{R}^n$ with coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a constant Poisson bracket on this cone. There is a version of this formalism applicable to $\mathbb{C}^n$ viewed as a real Poisson manifold. In this case, the tropicalization gives rise to a completely integrable system with action variables taking values in a polyhedral cone and angle variables spanning a torus. As an example, we consider the canonical Poisson bracket on the dual Poisson-Lie group $G^*$ for $G=U(n)$ in the cluster coordinates of Fomin-Zelevinsky defined by a certain choice of solid minors. We prove that the corresponding integrable system is isomorphic to the Gelfand-Zeitlin completely integrable system of Guillemin-Sternberg and Flaschka-Ratiu.