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Matrix Factorization for Solutions of the Yang–Baxter Equation

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We study solutions of the Yang–Baxter equation on the tensor product of an arbitrary finite-dimensional and an arbitrary infinite-dimensional representations of rank 1 symmetry algebra. We consider the cases of the Lie algebra sℓ2, the modular double (trigonometric deformation), and the Sklyanin algebra (elliptic deformation). The solutions are matrices with operator entries. The matrix elements are differential operators in the case of sℓ2, finite-difference operators with trigonometric coefficients in the case of the modular double, or finite-difference operators with coefficients constructed of the Jacobi theta functions in the case of the Sklyanin algebra. We find a new factorized form of the rational, trigonometric, and elliptic solutions, which drastically simplifies them. We show that they are products of several simply organized matrices and obtain for them explicit formulas. Bibliography: 44 titles.

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Authors and Affiliations

  1. St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia

    S. E. Derkachov

  2. Laboratoire d’Annecy-le-Vieux de Physique Théorique LAPTH, CNRS, UMR 5108 associée à l’Université de Savoie, Annecy-le-Vieux, France

    D. Chicherin

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  1. S. E. Derkachov
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Correspondence to S. E. Derkachov.

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Dedicated to Petr Kulish on the occasion of his 70th birthday

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 433, 2015, pp. 156–185.

Translated by S. E. Derkachov and D. I. Chicherin.

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Derkachov, S.E., Chicherin, D. Matrix Factorization for Solutions of the Yang–Baxter Equation. J Math Sci 213, 723–742 (2016). https://doi.org/10.1007/s10958-016-2734-0

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