Classical elliptic ${\mathrm{BC}}_1$ Ruijsenaars-van Diejen model: relation to Zhukovsky-Volterra gyrostat and 1-site classical XYZ model with boundaries

Abstract

We present a description of the classical elliptic ${\mathrm{BC}}_1$ Ruijsenaars-van Diejen model with 8 independent coupling constants through a pair of ${\mathrm{BC}}_1$ type classical Sklyanin algebras generated by the (classical) quadratic reflection equation with non-dynamical XYZ $r$-matrix. For this purpose, we consider the classical version of the $L$-operator for the Ruijsenaars-van Diejen model proposed by O. Chalykh. In ${\mathrm{BC}}_1$ case it is factorized to the product of two Lax matrices depending on 4 constants. Then we apply an IRF-Vertex type gauge transformation and obtain a product of the Lax matrices for the Zhukovsky-Volterra gyrostats. Each of them is described by the ${\mathrm{BC}}_1$ version of the classical Sklyanin algebra. In particular case, when 4 pairs of constants coincide, the ${\mathrm{BC}}_1$ Ruijsenaars-van Diejen model exactly coincides with the relativistic Zhukovsky-Volterra gyrostat. Explicit change of variables is obtained. We also consider another special case of the ${\mathrm{BC}}_1$ Ruijsenaars-van Diejen model with 7 independent constants. We show that it can be reproduced by considering the transfer matrix of the classical 1-site XYZ chain with boundaries. In the end of the paper, using another gauge transformation we represent the Chalykh's Lax matrix in a form depending on the Sklyanin's generators.

30 pages, typos corrected

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Cited by in corpus (1)

• Elliptic Ruijsenaars-Toda and elliptic Toda chains: classical r-matrix structure and relation to XYZ chain