Abstract
In this paper, we show that the Ginzburg–Weinstein diffeomorphism 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 . The target of the limit map is a product , where is the interior of a cone, T is a torus, and carries an integrable system with natural action-angle coordinates. The pull-back of these coordinates to 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’.
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