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

TL;DR

The paper develops a complete classical treatment of the elliptic BC$_1$ Ruijsenaars-van Diejen model with 8 couplings, revealing its underlying BC$_1$ Sklyanin-algebra structure via a classical reflection equation with a non-dynamical XYZ $r$-matrix. It constructs and factorizes the Lax matrix using Chalykh's formulation, then employs an IRF-Vertex gauge transformation to obtain a pair of Zhukovsky-Volterra gyrostats on a common phase space, establishing a Poisson map to the Sklyanin generators. Special cases are analyzed: a 4-constant submodel gauge-equivalent to the relativistic ZhV gyrostat, and a 1-site XYZ model with boundaries whose transfer-matrix recovers the RVd Hamiltonian in the appropriate limit. A second gauge transformation expresses Chalykh's Lax in terms of the original Sklyanin generators, and the Appendix provides the elliptic-function toolkit underpinning all constructions. Overall, the work connects relativistic and non-relativistic elliptic integrable systems through explicit Lax representations, gauge equivalences, and boundary-transfer methods, expanding the toolbox for classical and quantum Sklyanin-type algebras and their physical realizations.

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.

Theorems & Definitions (11)

• Lemma 2.1
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Theorem 4.1
• Lemma 4.1
• Theorem 4.2