Inequalities from Poisson brackets

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 $G^1$ for $G=U\left(n\right)$ 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.