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Noncommutative Inverse Scattering Method for the Kontsevich System

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We formulate an analog of Inverse Scattering Method for integrable systems on noncommutative associative algebras. In particular, we define Hamilton flows, Casimir elements and noncommutative analog of the Lax matrix. The noncommutative Lax element generates infinite family of commuting Hamilton flows on an associative algebra. The proposed approach to integrable systems on associative algebras satisfies certain universal property, in particular, it incorporates both classical and quantum integrable systems as well as provides a basis for further generalization. We motivate our definition by explicit construction of noncommutative analog of Lax matrix for a system of differential equations on associative algebra recently proposed by Kontsevich. First, we present these equations in the Hamilton form by defining a bracket of Loday type on the group algebra of the free group with two generators. To make the definition more constructive, we utilize (with certain generalizations) the Van den Bergh approach to Loday brackets via double Poisson brackets. We show that there exists an infinite family of commuting flows generated by the noncommutative Lax element.

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  1. Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd., Piscataway, 08854, NJ, USA

    Semeon Arthamonov

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  1. Semeon Arthamonov
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Arthamonov, S. Noncommutative Inverse Scattering Method for the Kontsevich System. Lett Math Phys 105, 1223–1251 (2015). https://doi.org/10.1007/s11005-015-0779-5

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  • DOI: https://doi.org/10.1007/s11005-015-0779-5

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