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Elliptic hydrodynamics and quadratic algebras of vector fields on a torus

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Abstract

We construct a quadratic Poisson algebra of Hamiltonian functions on a two-dimensional torus compatible with the canonical Poisson structure. This algebra is an infinite-dimensional generalization of the classical Sklyanin-Feigin-Odesskii algebras. It yields an integrable modification of the two-dimensional hydrodynamics of an ideal fluid on the torus. The Hamiltonian of the standard two-dimensional hydrodynamics is defined by the Laplace operator and thus depends on the metric. We replace the Laplace operator with a pseudodifferential elliptic operator depending on the complex structure. The new Hamiltonian becomes a member of a commutative bi-Hamiltonian hierarchy. In conclusion, we construct a Lie bialgebroid of vector fields on the torus.

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Authors and Affiliations

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

    M. A. Olshanetsky

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  1. M. A. Olshanetsky
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 355–370, March, 2007.

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Olshanetsky, M.A. Elliptic hydrodynamics and quadratic algebras of vector fields on a torus. Theor Math Phys 150, 301–314 (2007). https://doi.org/10.1007/s11232-007-0023-2

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  • DOI: https://doi.org/10.1007/s11232-007-0023-2

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