Inequalities from Poisson brackets

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 $G^{\ast }$ 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.

31 pages, 13 figures

Cited by in corpus (1)

• The $U(n)$ Gelfand-Zeitlin system as a tropical limit of Ginzburg-Weinstein diffeomorphisms