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 TMF, 1997, Volume 111, Number 2, Pages 182–217 (Mi tmf1001)

$R$-matrix quantization of the elliptic Ruijsenaars–Schneider model

G. E. Arutyunov, S. A. Frolov, L. O. Chekhov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/tmf1001

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English version:
Theoretical and Mathematical Physics, 1997, 111:2, 536–562

Bibliographic databases:

Citation: G. E. Arutyunov, S. A. Frolov, L. O. Chekhov, “$R$-matrix quantization of the elliptic Ruijsenaars–Schneider model”, TMF, 111:2 (1997), 182–217; Theoret. and Math. Phys., 111:2 (1997), 536–562

Citation in format AMSBIB
\Bibitem{AruFroChe97} \by G.~E.~Arutyunov, S.~A.~Frolov, L.~O.~Chekhov \paper $R$-matrix quantization of the elliptic Ruijsenaars--Schneider model \jour TMF \yr 1997 \vol 111 \issue 2 \pages 182--217 \mathnet{http://mi.mathnet.ru/tmf1001} \crossref{https://doi.org/10.4213/tmf1001} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1478891} \zmath{https://zbmath.org/?q=an:0978.81509} \elib{https://elibrary.ru/item.asp?id=13266180} \transl \jour Theoret. and Math. Phys. \yr 1997 \vol 111 \issue 2 \pages 536--562 \crossref{https://doi.org/10.1007/BF02634266} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1997YA85100003} 

• http://mi.mathnet.ru/eng/tmf1001
• https://doi.org/10.4213/tmf1001
• http://mi.mathnet.ru/eng/tmf/v111/i2/p182

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This publication is cited in the following articles:
1. Calogero, F, “Solution of certain integrable dynamical systems of Ruijsenaars-Schneider type with completely periodic trajectories”, Annales Henri Poincare, 1:1 (2000), 173
2. Lamers J., “Integral Formula For Elliptic Sos Models With Domain Walls and a Reflecting End”, Nucl. Phys. B, 901 (2015), 556–583
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