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

Relativistic classical integrable tops and quantum $R$-matrices

A. Levinab, M. Olshanetskycb, A. Zotovcdb

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 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.

 Funding Agency Grant Number Russian Foundation for Basic Research 12-02-0059412-01-00482 Ministry of Education and Science of the Russian Federation 11.G34.31.0023 Dynasty Foundation Russian Academy of Sciences - Federal Agency for Scientific Organizations Ï19 The work was supported by RFBR grants 12-02-00594 (A.L. and M.O.) and 12-01-00482 (A.Z.). The work of A.L. was partially supported by AG Laboratory GU-HSE, RF government grant, ag. 11 11.G34.31.0023. The work of A.Z. was partially supported by the D. Zimin's fund “Dynasty” and by the Program of RAS “Basic Problems of the Nonlinear Dynamics in Mathematical and Physical Sciences” Π19.

DOI: https://doi.org/10.1007/JHEP07(2014)012

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Accepted:10.06.2014
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