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

TL;DR

This paper explores the classical elliptic BC1 Ruijsenaars-van Diejen model, relating it to Zhukovsky-Volterra gyrostats and the classical XYZ model with boundaries, using Sklyanin algebras and gauge transformations.

Contribution

It provides a new algebraic description of the model via Sklyanin algebras and connects it explicitly to known integrable systems like the Zhukovsky-Volterra gyrostat and XYZ chain.

Findings

Model described by classical Sklyanin algebras

Special cases coincide with known integrable systems

Explicit variable changes and gauge transformations provided

Abstract

We present a description of the classical elliptic ${\rm BC}_1$ Ruijsenaars-van Diejen model with 8 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 ${\rm BC}_1$ 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 ${\rm BC}_1$ version of the classical Sklyanin algebra. In particular case, when 4 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 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.