
JHEP, 2014, Issue 7, 012, 39 pages
(Mi jhep11)




Relativistic classical integrable tops and quantum $R$matrices
A. Levin^{ab}, M. Olshanetsky^{cb}, A. Zotov^{cdb} ^{a} NRU HSE, Department of Mathematics,
Myasnitskaya str. 20, Moscow, 101000, Russia
^{b} ITEP,
B. Cheremushkinskaya str. 25, Moscow, 117218, Russia
^{c} MIPT,
Inststitutskii per. 9, Dolgoprudny, Moscow region, 141700, Russia
^{d} Steklov Mathematical Institute RAS,
Gubkina str. 8, Moscow, 119991, Russia
Abstract:
We describe classical toplike integrable systems arising from the quantum
exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed
in terms of the quantum nondynamical $R$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
nondynamical $R$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.
DOI:
https://doi.org/10.1007/JHEP07(2014)012
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Article Received: 04.06.2014 Accepted:10.06.2014
Language: English
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