Why are the rational and hyperbolic Ruijsenaars-Schneider hierarchies governed by the same R-operators as the Calogero-Moser ones?

Abstract

We demonstrate that in a certain gauge the Lax matrices of the rational and hyperbolic Ruijsenaars--Schneider models have a quadratic $r$-matrix Poisson bracket which is an exact quadratization of the linear $r$--matrix Poisson bracket of the Calogero--Moser models. This phenomenon is explained by a geometric derivation of Lax equations for arbitrary flows of both hierarchies, which turn out to be governed by the same dynamical $R$--operator.

LaTeX, 18pp, a revised version

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• Relativistic Calogero-Moser model as gauged WZW theory
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Cited by in corpus (20)

• Quantum vs Classical Integrability in Calogero-Moser Systems
• On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system
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• Explicit solutions of the classical Calogero & Sutherland systems for any root system
• Integrable discretizations of the spin Ruijsenaars-Schneider models
• Rational Calogero-Moser Model: Explicit Form and r-Matrix of the Second Poisson Structure
• The non-dynamical r-matrices of the degenerate Calogero-Moser models
• The Dn Ruijsenaars-Schneider model
• Poisson Structures of Calogero-Moser and Ruijsenaars-Schneider Models
• Quantum Trace Formulae for the Integrals of the Hyperbolic Ruijsenaars-Schneider model
• The dynamical twisting and nondynamical r-matrix structure of elliptic Ruijsenaars-Schneider model
• The Lax pairs for elliptic C_n and BC_n Ruijsenaars-Schneider models and their spectral curves
• Reduction of a bi-Hamiltonian hierarchy on $T^{\ast }\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models
• On the classical r-matrix structure of the rational BC(n) Ruijsenaars-Schneider-van Diejen system
• On the r-matrix structure of the hyperbolic BC(n) Sutherland model
• Classical r-matrix structure for elliptic Ruijsenaars chain and 1+1 field analogue of Ruijsenaars-Schneider model
• Quantum Dynamical $\check{R}$- Matrix with Spectral Parameter from Fusion
• On Calogero-Francoise-type Lax matrices and their dynamical r-matrices