Geodesic
$R$-matrix quantization of the elliptic Ruijsenaars–Schneider model
Teoretičeskaâ i matematičeskaâ fizika, Tome 111 (1997) no. 2, pp. 182-217 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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{\textendash}Schneider} model},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {182--217},
     year = {1997},
     volume = {111},
     number = {2},
     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
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
%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/

[1] O. Babelon, C. M. Viallet, Phys. Lett. B, 237 (1990), 411 | DOI | MR

[2] J. Avan, M. Talon, Phys. Lett. B, 303 (1993), 33–37 | DOI | MR

[3] Zh. Avan, O. Babelon, M. Talon, Algebra i analiz, 6:2 (1994), 67–89 | MR

[4] E. K. Sklyanin, Algebra i analiz, 6:2 (1994), 227–237 | MR | Zbl

[5] H. W. Braden, T. Suzuki, Lett. Math. Phys., 30 (1994), 147 | DOI | MR | Zbl

[6] J. Avan, G. Rollet, The classical $r$-matrix for the relativistic Ruijsenaars–Schneider system, Preprint BROWN-HET-1014, 1995 | MR

[7] Yu. B. Suris, Why are the rational and hyperbolic Ruijsenaars–Schneider hierarchies governed by the same $R$-matrix as the Calogero–Moser ones?, E-print hep-th/9602160 | MR

[8] F. W. Nijhoff, V. B. Kuznetsov, E. K. Sklyanin, O. Ragnisco, Dynamical $r$-matrix for the elliptic Ruijsenaars–Schneider model, E-print solv-int/9603006

[9] Yu. B. Suris, Elliptic Ruijsenaars–Schneider and Calogero–Moser hierarchies are governed by the same $r$-matrix, E-print solv-int/9603011

[10] S. N. Ruijsenaars, Commun. Math. Phys., 110 (1987), 191 | DOI | MR | Zbl

[11] M. A. Olshanetsky, A. M. Perelomov, Phys. Rep., 71 (1981), 313 | DOI | MR

[12] A. Gorsky, N. Nekrasov, Nucl. Phys. B, 414 (1994), 213 ; 436 (1995), 582 ; A. Gorsky, Integrable many body systems in the field theories, Prepint UUITP-16/94, 1994 | DOI | MR | Zbl | DOI | MR | Zbl

[13] G. E. Arutyunov, P. B. Medvedev, Phys. Lett. A, 223 (1996), 66–74 | DOI | MR | Zbl

[14] J. Avan, O. Babelon, E. Billey, The Gervais–Neveu–Felder equation and the quantum Calogero–Moser systems, Preprint PAR LPTHE 95-25, May 1995 | MR

[15] G. Felder, Conformal field theory and integrable systems associated to elliptic curves, E-print hep-th/9407154 | MR

[16] J. L. Gervais, A. Neveu, Nucl. Phys. B, 238 (1984), 125 | DOI | MR

[17] O. Babelon, D. Bernard, E. Billey, A quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and difference equations, Preprint PAR LPTHE 95-51 | MR

[18] G. E. Arutyunov, S. A. Frolov, Quantum dynamical $R$-matrices and quantum Frobenius group, E-print q-alg/9610009 | MR

[19] G. E. Arutyunov, S. A. Frolov, P. B. Medvedev, Elliptic Ruijsenaars–Schneider model via the Poisson reduction of the affine Heisenberg double, E-print hep-th/9607170 | MR

[20] G. E. Arutyunov, S. A. Frolov, P. B. Medvedev, Elliptic Ruijsenaars–Schneider model from the cotangent bundle over the two-dimensional current group, E-print hep-th/9608013 | MR

[21] K. Hasegawa, Ruijsenaars' commuting difference operators as commuting transfer matrices, E-print q-alg/9512029 | MR

[22] A. A. Belavin, Nucl. Phys. B, 180 (1981), 189–200 | DOI | MR | Zbl

[23] P. I. Etingof, I. B. Frenkel, Central extensions of current groups in two dimensions, E-print hep-th/9303047 | MR

[24] F. Falceto, K. Gawedski, Commun. Math. Phys., 159 (1994), 549 | DOI | MR | Zbl

[25] L. D. Faddeev, “Integrable models in $(1+1)$-dimensional quantum field theory”, Recent advances in field theory and statistical mechanics, Les Houches Summer School Proc. Session XXXIX (1982), eds. J. B. Zuber, R. Stora, Elsvier Sci. Publ., 1984, 561 | MR

[26] P. P. Kulish, E. K. Sklyanin, “Quantum spectral transform method. Recent developments”, Integrable quantum field theories, eds. J. Hietarinta, C. Montonen; Lect. Notes. Phys., 51, 1982, 61 | DOI

[27] M. P. Richey, C. A. Tracy, J. Stat. Phys., 42 (1986), 311–348 | DOI | MR

[28] Y. Quano, A. Fujii, Mod. Phys. Lett. A, 8:17 (1993), 1585–1597 | DOI | MR | Zbl

[29] K. Hasegawa, J. Phys. A, 26 (1993), 3211–3228 | DOI | MR | Zbl

[30] R. J. Baxter, Ann. Phys., 76 (1973), 1–24 ; 25–47 ; 48–71 | DOI | Zbl | Zbl | Zbl

[31] M. Jimbo, T. Miwa, M. Okado, Lett. Math. Phys., 14 (1987), 123–131 ; Nucl. Phys. B, 300 (1988), 74–108 | DOI | MR | Zbl | DOI | MR

[32] K. Hasegawa, J. Math. Phys., 35:11 (1994), 6158–6171 | DOI | MR | Zbl

[33] E. K. Sklyanin, Funkts. analiz i ego prilozh., 16:4 (1982), 27–34 ; 17:4, 34–48 | MR | Zbl | MR | Zbl

[34] I. M. Krichever, O. Babelon, E. Billey, M. Talon, Spin generalization of Calogero–Moser system and the matrix KP equation, E-print hep-th/9411160 | MR

[35] N. Nekrasov, Holomorphic bundles and many-body systems, E-print hep-th/9503157 | MR

[36] B. Enriquez, V. Rubtsov, Math. Research Lett., 3:3 (1996), 343–358 | DOI | MR

[37] I. M. Krichever, A. V. Zabrodin, Spin generalizations of the Ruijsenaars–Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra, E-print hep-th/9505039 | MR

[38] V. B. Kuznetsov, E. K. Sklyanin, Separation of variables for $A_2$ Ruijsenaars model and new integral representation for $A_2$ Macdonald polinomials, E-print q-alg/9602023 | MR