Elliptic Ruijsenaars-Schneider and Calogero-Moser hierarchies are governed by the same r-matrix

Abstract

We demonstrate that in a certain gauge the elliptic Ruijsenaars--Schneider models admit Lax representation governed by the same dynamical $r$--matrix as their non--relativistic counterparts (Calogero--Moser models). This phenomenon was previously observed for the rational and hyperbolic models.

References in corpus (6)

• Relativistic Calogero-Moser model as gauged WZW theory
• The Sine-Gordon Solitons as a N-Body Problem
• The R-matrix structure of the Euler-Calogero-Moser model
• Exact Yangian Symmetry in the Classical Euler-Calogero-Moser Model
• Dynamical r-matrix for the elliptic Ruijsenaars-Schneider system
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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
• Construction of dynamical quadratic algebras
• From Hamiltonian to zero curvature formulation for classical integrable boundary conditions
• Integrability of the $C_n$ and $B{C}_n$ Ruijsenaars-Schneider models
• 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