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Elliptic Spectral Parameter and Infinite-Dimensional Grassmann Variety

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Infinite Dimensional Algebras and Quantum Integrable Systems

Part of the book series: Progress in Mathematics ((PM,volume 237))

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Abstract

Recent results on the Grassmannian perspective of soliton equations with an elliptic spectral parameter are presented along with a detailed review of the classical case with a rational spectral parameter. The nonlinear Schrödinger hierarchy is picked out for illustration of the classical case. This system is formulated as a dynamical system on a Lie group of Laurent series with factorization structure. The factorization structure induces a mapping to an infinite-dimensional Grassmann variety. The dynamical system on the Lie group is thereby mapped to a simple dynamical system on a subset of the Grassmann variety. Upon suitable modification, almost the same procedure turns out to work for soliton equations with an elliptic spectral parameter. A clue is the geometry of holomorphic vector bundles over the elliptic curve hidden (or manifest) in the zero-curvature representation.

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Authors and Affiliations

  1. Graduate School of Human and Environmental Studies, Kyoto University, Yoshida Sakyo, Kyoto, 606-8501, Japan

    Kanehisa Takasaki

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  1. Kanehisa Takasaki
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Editors and Affiliations

  1. St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontaka 27, 191011, St. Petersburg, Russia

    Petr P. Kulish

  2. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005-139, Faro, Portugal

    Nenad Manojlovich

  3. IInd Institute for Theoretical Physics, University of Hamburg, Luruper Chaussee 149, 22761, Hamburg, Germany

    Henning Samtleben

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Takasaki, K. (2005). Elliptic Spectral Parameter and Infinite-Dimensional Grassmann Variety. In: Kulish, P.P., Manojlovich, N., Samtleben, H. (eds) Infinite Dimensional Algebras and Quantum Integrable Systems. Progress in Mathematics, vol 237. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7341-5_6

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