$R$-matrix quantization of the elliptic Ruijsenaars--Schneider model
Teoretičeskaâ i matematičeskaâ fizika, Tome 111 (1997) no. 2, pp. 182-217
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It is shown that the classical $L$-operator algebra of the elliptic Ruijsenaars–Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamical $r$ and $\bar r$-matrices satisfying a closed system of equations. The corresponding quantum $R$ and $\overline R$-matrices are found as solutions to quantum analogs of these equations. We present the quantum $L$-operator algebra and show that the system of equations for $R$ and $\overline R$ arises as the compatibility condition for this algebra. It turns out that the $R$-matrix is twist-equivalent to the Felder elliptic $R^F$-matrix with $\overline R$ playing the role of the twist. The simplest representation of the quantum $L$-operator algebra corresponding to the elliptic Ruijsenaars–Schneider model is obtained. The connection of the quantum $L$- operator algebra to the fundamental relation $RLL=LLR$ with Belavin's elliptic $R$-matrix is established. Asa byproduct of our construction, we find a new $N$-parameter elliptic solution to the classical Yang–Baxter equation.
@article{TMF_1997_111_2_a2,
author = {G. E. Arutyunov and S. A. Frolov and L. O. Chekhov},
title = {$R$-matrix quantization of the elliptic {Ruijsenaars--Schneider} model},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {182--217},
publisher = {mathdoc},
volume = {111},
number = {2},
year = {1997},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_1997_111_2_a2/}
}
TY - JOUR AU - G. E. Arutyunov AU - S. A. Frolov AU - L. O. Chekhov TI - $R$-matrix quantization of the elliptic Ruijsenaars--Schneider model JO - Teoretičeskaâ i matematičeskaâ fizika PY - 1997 SP - 182 EP - 217 VL - 111 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_1997_111_2_a2/ LA - ru ID - TMF_1997_111_2_a2 ER -
%0 Journal Article %A G. E. Arutyunov %A S. A. Frolov %A L. O. Chekhov %T $R$-matrix quantization of the elliptic Ruijsenaars--Schneider model %J Teoretičeskaâ i matematičeskaâ fizika %D 1997 %P 182-217 %V 111 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_1997_111_2_a2/ %G ru %F TMF_1997_111_2_a2
G. E. Arutyunov; S. A. Frolov; L. O. Chekhov. $R$-matrix quantization of the elliptic Ruijsenaars--Schneider model. Teoretičeskaâ i matematičeskaâ fizika, Tome 111 (1997) no. 2, pp. 182-217. http://geodesic.mathdoc.fr/item/TMF_1997_111_2_a2/