Relativistic Classical Integrable Tops and Quantum R-matrices

Abstract

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 ${\mathrm{gl}}_N$ 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 ${\mathrm{gl}}_N$ 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 ${\mathrm{gl}}_2$ 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.

36 pages, minor corrections

References in corpus (6)

• Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics
• Classical integrable systems and soliton equations related to eleven-vertex R-matrix
• Characteristic Classes and Integrable Systems for Simple Lie Groups
• Characteristic Classes and Integrable Systems. General Construction
• Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
• Vector bundles on degenerations of elliptic curves and Yang-Baxter equations

Cited by in corpus (4)

• Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations
• Classical integrable systems and soliton equations related to eleven-vertex R-matrix
• Relativistic elliptic matrix tops and finite Fourier transformations
• Nonstandard representations of type C affine Hecke algebra from K-operators