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The

A Alekseev1, J Lane2, Y Li2

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

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
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Summary
This summary is machine-generated.

The Ginzburg-Weinstein diffeomorphism has a scaling tropical limit, revealing an integrable system. This work connects geometric structures with integrable systems, offering new insights into their properties.

Keywords:
Gelfand–ZeitlinPoisson geometryPoisson–Lie groupsintegrable systemstropicalization

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Area of Science:

  • Differential Geometry
  • Mathematical Physics
  • Integrable Systems

Background:

  • The Ginzburg-Weinstein diffeomorphism is a key object in symplectic geometry.
  • Integrable systems are fundamental in classical and quantum mechanics.
  • The Gelfand-Zeitlin and Flaschka-Ratiu systems are important examples of integrable systems.

Purpose of the Study:

  • To investigate the scaling tropical limit of the Ginzburg-Weinstein diffeomorphism.
  • To identify the structure of the limit space and its associated integrable system.
  • To explore connections between different integrable systems and geometric structures.

Main Methods:

  • Analysis of the Ginzburg-Weinstein diffeomorphism.
  • Construction of a scaling tropical limit map.
  • Identification of action-angle coordinates on the limit space.
  • Investigation of Lagrangian tori within integrable systems.

Main Results:

  • The Ginzburg-Weinstein diffeomorphism admits a scaling tropical limit.
  • The limit space is a product of a cone interior, a torus, and a space carrying an integrable system.
  • The pull-back of coordinates recovers the Gelfand-Zeitlin integrable system.
  • Lagrangian tori of the Flaschka-Ratiu system intersect totally positive matrices.

Conclusions:

  • The study establishes a novel connection between geometric diffeomorphisms and integrable systems through tropical limits.
  • The findings provide a new perspective on the Gelfand-Zeitlin and Flaschka-Ratiu integrable systems.
  • This research contributes to the understanding of finite-dimensional integrable systems and their underlying geometric structures.