?
Relativistic Classical Integrable Tops and Quantum R-matrices
Cornell University
,
2014.
e 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.