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The relative frame bundle of an infinite-dimensional flag variety and solutions of integrable hierarchies

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Abstract

We develop a group theory approach for constructing solutions of integrable hierarchies corresponding to the deformation of a collection of commuting directions inside the Lie algebra of upper-triangular ZxZ matrices. Depending on the choice of the set of commuting directions, the homogeneous space from which these solutions are constructed is the relative frame bundle of an infinite-dimensional flag variety or the infinite-dimensional flag variety itself. We give the evolution equations for the perturbations of the basic directions in the Lax form, and they reduce to a tower of differential and difference equations for the coefficients of these perturbed matrices. The Lax equations follow from the linearization of the hierarchy and require introducing a proper analogue of the Baker—Akhiezer function.

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Authors and Affiliations

  1. Korteweg—de Vries Institute of Mathematics, University of Amsterdam, Amsterdam, The Netherlands

    G. F. Helminck

  2. North Carolina State University, Raleigh, North Carolina, USA

    A. G. Helminck

  3. Orenburg State Pedagogical University, Orenburg, Russia

    A. V. Opimakh

Authors
  1. G. F. Helminck
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  2. A. G. Helminck
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  3. A. V. Opimakh
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Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 165, No. 3, pp. 440–471, December, 2010.

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Helminck, G.F., Helminck, A.G. & Opimakh, A.V. The relative frame bundle of an infinite-dimensional flag variety and solutions of integrable hierarchies. Theor Math Phys 165, 1610–1636 (2010). https://doi.org/10.1007/s11232-010-0133-0

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  • DOI: https://doi.org/10.1007/s11232-010-0133-0

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