It is shown that the sl 2 trigonometric Ruijsenaars-Schneider model admits a nondynamical r -matrix structure under a certain gauge. The corresponding r -matrix is the classical limit of a twisted trigonometric R -matrix. The new factorized classical and quantum L -operators are obtained.
We extend the quantum–classical duality to the trigonometric (hyperbolic) case. The duality establishes an explicit relationship between the classical N-body trigonometric Ruijsenaars–Schneider model and the inhomogeneous twisted XXZ spin chain on N sites. Similarly to the rational version, the spin chain data fixes a certain Lagrangian submanifold in the phase space of the classical integrable system. The inhomogeneity parameters are equal to the coordinates of particles while the velocities of classical particles are proportional to the eigenvalues of the spin chain Hamiltonians (residues of the properly normalized transfer matrix). In the rational version of the duality, the action variables of the Ruijsenaars–Schneider model are equal to the twist parameters with some multiplicities defined by quantum (occupation) numbers. In contrast to the rational version, in the trigonometric case there is a splitting of the spectrum of action variables (eigenvalues of the classical Lax matrix). The limit corresponding to the classical Calogero–Sutherland system and quantum trigonometric Gaudin model is also described as well as the XX limit to free fermions.
We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical R-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable systems (relativistic tops) are described as multidimensional Euler tops, and the inertia tensors are written in terms of the quantum and classical R-matrices. A particular case of gl N system is gauge equivalent to the N-particle RS model while a generic top is related to the spin generalization of the RS model. The simple relation between quantum R-matrices and classical Lax operators is exploited in two ways. In the elliptic case we use the Belavin’s quantum R-matrix to describe the relativistic classical tops. Also by the passage to the noncommutative torus we study the large N limit corresponding to the relativistic version of the nonlocal 2d elliptic hydrodynamics. Conversely, in the rational case we obtain a new gl N quantum rational non-dynamical R-matrix via the relativistic top, which we get in a different way — using the factorized form of the RS Lax operator and the classical Symplectic Hecke (gauge) transformation. In particular case of gl2 the quantum rational R-matrix is 11-vertex. It was previously found by Cherednik. At last, we describe the integrable spin chains and Gaudin models related to the obtained R-matrix.