Alekseev A, Lane J, Li Y
Department of Mathematics, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland
Department of Mathematics, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland.
Philos Trans A Math Phys Eng Sci. 2018 Sep 17;376(2131):20170428. doi: 10.1098/rsta.2017.0428.
In this paper, we show that the Ginzburg-Weinstein diffeomorphism [Formula: see text] of Alekseev & Meinrenken (Alekseev, Meinrenken 2007 , 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, 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 , 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'.
在本文中，我们证明了Alekseev和Meinrenken（Alekseev, Meinrenken 2007, 1 - 34. (10.4310/jdg/1180135664)）的金兹堡 - 温斯坦微分同胚[公式：见原文]在[公式：见原文]的一个开稠密子集上允许一个缩放热带极限。极限映射的目标是一个乘积[公式：见原文]，其中[公式：见原文]是一个锥的内部，[公式：见原文]是一个环面，并且[公式：见原文]带有一个具有自然作用 - 角坐标的可积系统。将这些坐标拉回到[公式：见原文]可恢复Guillemin和Sternberg（Guillemin, Sternberg 1983, 106 - 128. (10.1016/0022 - 1236(83)90092 - 7)）的盖尔范德 - 蔡特林可积系统。作为我们证明的一个副产品，我们表明对于足够大的作用坐标，上三角矩阵集上的弗拉施卡 - 拉蒂乌可积系统的拉格朗日环面与全正矩阵集相交。本文是主题为“有限维可积系统：新趋势与方法”的一部分。
