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Poisson-Lie Interpretation of Trigonometric Ruijsenaars Duality

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Abstract

A geometric interpretation of the duality between two real forms of the complex trigonometric Ruijsenaars-Schneider system is presented. The phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard Heisenberg double of U(n). The collections of commuting Hamiltonians of the systems in duality are shown to descend from two families of ‘free’ Hamiltonians on the double which are dual to each other in a Poisson-Lie sense. Our results give rise to a major simplification of Ruijsenaars’ proof of the crucial symplectomorphism property of the duality map.

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Authors and Affiliations

  1. Department of Theoretical Physics, MTA KFKI RMKI, P.O.B. 49, 1525, Budapest, Hungary

    L. Fehér

  2. Department of Theoretical Physics, University of Szeged, Tisza Lajos krt 84-86, H-6720, Szeged, Hungary

    L. Fehér

  3. Institut de Mathématiques de Luminy, 163, Avenue de Luminy, 13288, Marseille, France

    C. Klimčí k

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  1. L. Fehér
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  2. C. Klimčí k
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Correspondence to L. Fehér.

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Communicated by M. Aizenman

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Fehér, L., Klimčí k, C. Poisson-Lie Interpretation of Trigonometric Ruijsenaars Duality. Commun. Math. Phys. 301, 55–104 (2011). https://doi.org/10.1007/s00220-010-1140-6

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