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Painlevé VI, Rigid Tops and Reflection Equation

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Abstract

We show that the Painlevé VI equation has an equivalent form of the non-autonomous Zhukovsky-Volterra gyrostat. This system is a generalization of the Euler top in \(\mathbb{C}^3\) and includes the additional constant gyrostat momentum. The quantization of its autonomous version is achieved by the reflection equation. The corresponding quadratic algebra generalizes the Sklyanin algebra. As by product we define integrable XYZ spin chain on a finite lattice with new boundary conditions.

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Authors and Affiliations

  1. Max Planck Institute of Mathematics, Bonn, Germany

    A. M. Levin & M. A. Olshanetsky

  2. Institute of Oceanology, Moscow, Russia

    A. M. Levin

  3. Institute of Theoretical and Experimental Physics, Moscow, Russia

    M. A. Olshanetsky & A. V. Zotov

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  1. A. M. Levin
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  2. M. A. Olshanetsky
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  3. A. V. Zotov
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Correspondence to M. A. Olshanetsky.

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Communicated by L. Takhtajan

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Levin, A.M., Olshanetsky, M.A. & Zotov, A.V. Painlevé VI, Rigid Tops and Reflection Equation. Commun. Math. Phys. 268, 67–103 (2006). https://doi.org/10.1007/s00220-006-0089-y

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