Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 079, 13 pages      arXiv:1207.5368

Rational Calogero-Moser Model: Explicit Form and r-Matrix of the Second Poisson Structure

Jean Avan a and Eric Ragoucy b
a) Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise (CNRS UMR 8089), Saint-Martin 2, 2, av. Adolphe Chauvin, F-95302 Cergy-Pontoise Cedex, France
b) LAPTH Annecy le Vieux, CNRS and Université de Savoie, 9 chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France

Received July 24, 2012, in final form October 17, 2012; Published online October 26, 2012

We compute the full expression of the second Poisson bracket structure for N=2 and N=3 site rational classical Calogero-Moser model. We propose an r-matrix formulation for N=2. It is identified with the classical limit of the second dynamical boundary algebra previously built by the authors.

Key words: classical integrable systems; hierarchy of Poisson structures; dynamical reflection equation.

pdf (372 kb)   tex (22 kb)


  1. Aniceto I., Avan J., Jevicki A., Poisson structures of Calogero-Moser and Ruijsenaars-Schneider models, J. Phys. A: Math. Gen. 43 (2010), 185201, 14 pages, arXiv:0912.3468.
  2. Arutyunov G.E., Chekhov L.O., Frolov S.A., R-matrix quantization of the elliptic Ruijsenaars-Schneider model, Comm. Math. Phys. 192 (1998), 405-432, q-alg/9612032.
  3. Arutyunov G.E., Frolov S.A., Quantum dynamical R-matrices and quantum Frobenius group, Comm. Math. Phys. 191 (1998), 15-29, q-alg/9610009.
  4. Avan J., Babelon O., Talon M., Construction of the classical R-matrices for the Toda and Calogero models, St. Petersburg Math. J. 6 (1994), 255-274, hep-th/9606102.
  5. Avan J., Billaud B., Rollet G., Classification of non-affine non-Hecke dynamical R-matrices, SIGMA 8 (2012), 064, 45 pages, arXiv:1204.2746.
  6. Avan J., Ragoucy E., A new dynamical reflection algebra and related quantum integrable systems, Lett. Math. Phys. 101 (2012), 85-101, arXiv:1106.3264.
  7. Avan J., Talon M., Classical R-matrix structure for the Calogero model, Phys. Lett. B 303 (1993), 33-37, hep-th/9210128.
  8. Babelon O., Bernard D., Talon M., Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003.
  9. Babelon O., Viallet C.M., Hamiltonian structures and Lax equations, Phys. Lett. B 237 (1990), 411-416.
  10. Balog J., D abrowski L., Fehér L., Classical r-matrix and exchange algebra in WZNW and Toda theories, Phys. Lett. B 244 (1990), 227-234.
  11. Bartocci C., Falqui G., Mencattini I., Ortenzi G., Pedroni M., On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system, Int. Math. Res. Not. 2010 (2010), no. 2, 279-296, arXiv:0902.0953.
  12. Braden H.W., Suzuki T., R-matrices for elliptic Calogero-Moser models, Lett. Math. Phys. 30 (1994), 147-158, hep-th/9312031.
  13. Calogero F., Exactly solvable one-dimensional many-body problems, Lett. Nuovo Cimento 13 (1975), 411-416.
  14. Calogero F., On a functional equation connected with integrable many-body problems, Lett. Nuovo Cimento 16 (1976), 77-80.
  15. Fehér L., Pusztai B.G., A class of Calogero type reductions of free motion on a simple Lie group, Lett. Math. Phys. 79 (2007), 263-277, math-ph/0609085.
  16. Felder G., Elliptic quantum groups, in XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 1995, 211-218, hep-th/9412207.
  17. Freidel L., Maillet J.M., Quadratic algebras and integrable systems, Phys. Lett. B 262 (1991), 278-284.
  18. Gervais J.L., Neveu A., Novel triangle relation and absence of tachyons in Liouville string field theory, Nuclear Phys. B 238 (1984), 125-141.
  19. Lax P.D., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490.
  20. Li L.C., Parmentier S., Nonlinear Poisson structures and r-matrices, Comm. Math. Phys. 125 (1989), 545-563.
  21. Magri F., Casati P., Falqui G., Pedroni M., Eight lectures on integrable systems, in Integrability of Nonlinear Systems (Pondicherry, 1996), Lecture Notes in Phys., Vol. 495, Springer, Berlin, 1997, 256-296.
  22. Maillet J.M., New integrable canonical structures in two-dimensional models, Nuclear Phys. B 269 (1986), 54-76.
  23. Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220.
  24. Nagy Z., Avan J., Rollet G., Construction of dynamical quadratic algebras, Lett. Math. Phys. 67 (2004), 1-11, math.QA/0307026.
  25. Oevel W., Ragnisco O., R-matrices and higher Poisson brackets for integrable systems, Phys. A 161 (1989), 181-220.
  26. Olshanetsky M.A., Perelomov A.M., Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), 313-400.
  27. Pusztai B.G., On the r-matrix structure of the hyperbolic BCn model, arXiv:1205.1029.
  28. Ruijsenaars S.N.M., Schneider H., A new class of integrable systems and its relation to solitons, Ann. Physics 170 (1986), 370-405.
  29. Semenov-Tjan-Shanskii M.A., What is a classical r-matrix?, Funct. Anal. Appl. 17 (1983), 259-272.
  30. Sklyanin E.K., Some algebraic structures connected with the Yang-Baxter equation, Funct. Anal. Appl. 16 (1982), 263-270.
  31. Suris Yu.B., Why is the Ruijsenaars-Schneider hierarchy governed by the same R-operator as the Calogero-Moser one?, Phys. Lett. A 225 (1997), 253-262, hep-th/9602160.
  32. Xu P., Quantum dynamical Yang-Baxter equation over a nonabelian base, Comm. Math. Phys. 226 (2002), 475-495, math.QA/0104071.

Previous article  Next article   Contents of Volume 8 (2012)