Liouville integrability of classical Calogero-Moser models. (English) Zbl 0972.81216

Summary: Liouville integrability of classical Calogero-Moser models is proved for models based on any root system, including the non-crystallographic ones. It applies to all types of elliptic potentials, i.e., untwisted and twisted together with their degenerations (hyperbolic, trigonometric and rational), except for the rational potential models confined by a harmonic force.


81V70 Many-body theory; quantum Hall effect
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
Full Text: DOI arXiv


[1] Calogero, F.; Sutherland, B.; Moser, J.; Moser, J.; Calogero, F.; Marchioro, C.; Ragnisco, O.; Calogero, F., Integrable systems of non-linear evolution equations, (), Lett. nuovo cimento, Lett. nuovo cimento, 13, 411, (1975)
[2] Olshanetsky, M.A.; Perelomov, A.M.; Olshanetsky, M.A.; Perelomov, A.M., Invent. math., Phys. rep., 94, 313, (1983)
[3] van Diejen, J.F.; Vinet, L., Calogero – moser – sutherland models, (2000), Springer
[4] Khastgir, S.P.; Pocklington, A.J.; Sasaki, R., J. phys. A, 33, 9033, (2000)
[5] Bordner, A.J.; Corrigan, E.; Sasaki, R., Prog. theor. phys., 102, 499, (1999)
[6] Bordner, A.J.; Corrigan, E.; Sasaki, R.; Bordner, A.J.; Sasaki, R.; Takasaki, K.; Bordner, A.J.; Sasaki, R.; Khastgir, S.P.; Sasaki, R.; Takasaki, K., Prog. theor. phys., Prog. theor. phys., Prog. theor. phys., Prog. theor. phys., 102, 749, (1999)
[7] D’Hoker, E.; Phong, D.H., Nucl. phys. B, 530, 537, (1998)
[8] Olshanetsky, M.A.; Perelomov, A.M., Funct. anal. appl., 12, 121, (1977)
[9] Hurtubise, J.C.; Markman, E., 1999
[10] Ochiai, H.; Oshima, T.; Sekiguchi, H.; Oshima, T.; Sekiguchi, H., Proc. jpn. acad. ser. A math. sci., J. math. sci. univ. Tokyo, 2, 1, (1995)
[11] Inozemtsev, V.I., Lett. math. phys., 17, 11, (1989)
[12] Krichever, I.M., Funct. anal. appl., 14, 282, (1980)
[13] Flatto, L.; Wiener, M.M.; Flatto, L., Amer. J. math., Amer. J. math., 92, 552, (1970)
[14] Humphreys, J.E., Reflection groups and Coxeter groups, (1990), Cambridge University Press Cambridge · Zbl 0725.20028
[15] Shioda, T., J. math. soc. jpn., 43, 673, (1991)
[16] Berezin, F.A., Trudy moskov. mat. obshch., 6, 371, (1957)
[17] Avan, J.; Talon, M.; Avan, J.; Babelon, O.; Talon, M., Phys. lett. B, Algebra anal., 6, 67, (1994)
[18] Sklyanin, E.K.; Braden, H.W.; Suzuki, T., Algebra anal., Lett. math. phys., 30, 147, (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.