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Inequalities from Poisson brackets | | Anton Alekseev
; Irina Davydenkova
; | | Date: |
13 May 2015 | | 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. | | Source: | arXiv, 1505.3233 | | Services: | Forum | Review | PDF | Favorites | |
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