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arXiv · 1505.03233

Inequalities from Poisson brackets

Anton Alekseev·Irina Davydenkova
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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.

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Anton Alekseev, Irina Davydenkova. 2015-05-13. Inequalities from Poisson brackets. https://doi.org/10.1016/j.indag.2014.07.003

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