Geodesic
$R$-matrix quantization of the elliptic Ruijsenaars–Schneider model
Teoretičeskaâ i matematičeskaâ fizika, Tome 111 (1997) no. 2, pp. 182-217 
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/
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     author = {G. E. Arutyunov and S. A. Frolov and L. O. Chekhov},
     title = {$R$-matrix quantization of the elliptic {Ruijsenaars{\textendash}Schneider} model},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {182--217},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1997_111_2_a2/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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