Relativistic Classical Integrable Tops and Quantum R-matrices

TL;DR

This paper introduces a new class of relativistic integrable systems called relativistic tops, derived from quantum R-matrices, and explores their connections to known models like the Ruijsenaars-Schneider system, including their classical and quantum properties.

Contribution

The paper constructs relativistic classical tops from quantum exchange relations and R-matrices, revealing new links between quantum R-matrices and classical integrable systems, including novel rational R-matrices.

Findings

Relativistic tops are described as multidimensional Euler tops with inertia tensors from R-matrices.

A new gl_N quantum rational R-matrix is derived, generalizing known models.

Connections between quantum R-matrices and classical Lax operators are established, including the 11-vertex R-matrix.

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