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

A Alekseev; J Lane; Y Li

doi:10.1098/rsta.2017.0428

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

Philos Trans A Math Phys Eng Sci. 2018 Sep 17;376(2131):20170428. doi: 10.1098/rsta.2017.0428.

Authors

A Alekseev¹, J Lane², Y Li²

Affiliations

¹ Department of Mathematics, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland anton.alekseev@unige.ch.
² Department of Mathematics, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland.

Abstract

In this paper, we show that the Ginzburg-Weinstein diffeomorphism [Formula: see text] of Alekseev & Meinrenken (Alekseev, Meinrenken 2007 J. Differential Geom.76, 1-34. (10.4310/jdg/1180135664)) admits a scaling tropical limit on an open dense subset of [Formula: see text] The target of the limit map is a product [Formula: see text], where [Formula: see text] is the interior of a cone, T is a torus, and [Formula: see text] carries an integrable system with natural action-angle coordinates. The pull-back of these coordinates to [Formula: see text] recovers the Gelfand-Zeitlin integrable system of Guillemin & Sternberg (Guillemin, Sternberg 1983 J. Funct. Anal.52, 106-128. (10.1016/0022-1236(83)90092-7)). As a by-product of our proof, we show that the Lagrangian tori of the Flaschka-Ratiu integrable system on the set of upper triangular matrices meet the set of totally positive matrices for sufficiently large action coordinates.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Keywords: Gelfand–Zeitlin; Poisson geometry; Poisson–Lie groups; integrable systems; tropicalization.