Classical elliptic $${\rm BC}_1$$ 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 $${\rm BC}_1$$ Ruijsenaars–van Diejen model with eight independent coupling constants through a pair of $${\rm BC}_1$$ 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 the $${\rm BC}_1$$ case, it is factorized into the product of two Lax matrices depending on four 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 $${\rm BC}_1$$ version of the classical Sklyanin algebra. In particular case, when four pairs of constants coincide, the $${\rm BC}_1$$ 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 $${\rm BC}_1$$ Ruijsenaars–van Diejen model with seven 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 Chalykh’s Lax matrix in a form depending on Sklyanin’s generators.