Spin Calogero models obtained from dynamical r-matrices and geodesic motion

Nuclear Physics B

Volumn 734, Issue 3, 2006, Pages 304-325

  Feher L ^a,b   Pusztai, B G ^c,d  

^a WIGNER RESEARCH CENTRE FOR PHYSICS   (Hungary)

^b UNIVERSITY OF SZEGED   (Hungary)

^c UNIVERSITÉ DE MONTRÉAL   (Canada)

^d Dept of Electrical and Computer Engineering   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 29744449373     PISSN: 05503213     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nuclphysb.2005.11.025     Document Type: Article

References (50)

1. F. Calogero Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials J. Math. Phys. 12 1971 419-436
2. B. Sutherland Exact results for a quantum many body problem in one dimension Phys. Rev. A 4 1971 2019-2021
3. J. Moser Three integrable Hamiltonian systems connected with isospectral deformations Adv. Math. 16 1975 197-220
4. A.M. Perelomov Integrable Systems of Classical Mechanics and Lie Algebras 1990 Birkhäuser Basel
5. N. Nekrasov Infinite-dimensional algebras, many-body systems and gauge theories in: A.Yu. Morozov M.A. Olshanetsky (Eds) Moscow Seminar in Mathematical Physics, AMS Transl. Ser., vol. 2 1999 American Mathematical Society Providence 263-299
6. J.F. van Diejen L. Vinet Calogero-Moser-Sutherland Models 2000 Spinger Berlin
7. B. Sutherland Beautiful Models 2004 World Scientific Singapore
8. M.A. Olshanetsky A.M. Perelomov Completely integrable Hamiltonian systems connected with semisimple Lie algebras Invent. Math. 37 1976 93-108
9. J. Gibbons T. Hermsen A generalisation of the Calogero-Moser system Physica D 11 1984 337-348
10. S.N.M. Ruijsenaars H. Schneider A new class of integrable models and their relation to solitons Ann. Phys. (N.Y.) 170 1986 370-405
11. M.A. Olshanetsky A.M. Perelomov Explicit solution of the Calogero model in the classical case and geodesic flows on symmetric spaces of zero curvature Lett. Nouvo Cimento 16 1976 333-339
12. M.A. Olshanetsky A.M. Perelomov Explicit solutions of some completely integrable systems Lett. Nuovo Cimento 17 1976 97-101
13. D. Kazhdan B. Kostant S. Sternberg Hamiltonian group actions and dynamical systems of Calogero type Commun. Pure Appl. Math. XXXI 1978 481-507
14. O. Babelon C.-M. Viallet Hamiltonian structures and Lax equations Phys. Lett. B 237 1990 411-416 hep-ph/9210128
15. J. Avan M. Talon Classical R-matrix structure for the Calogero model Phys. Lett. B 303 1993 33-37 hep-ph/9308060
16. E.K. Sklyanin Dynamical r-matrices for the elliptic Calogero-Moser model Algebra Anal. 6 1994 227-237 hep-ph/9309033
17. H.W. Braden T. Suzuki R-matrices for elliptic Calogero-Moser models Lett. Math. Phys. 30 1994 147-158 hep-ph/9312042
18. E. Billey J. Avan O. Babelon The r-matrix structure of the Euler-Calogero-Moser model Phys. Lett. A 186 1994 114-118 hep-ph/9505091
19. E. Billey J. Avan O. Babelon Exact Yangian symmetry in the classical Euler-Calogero-Moser model Phys. Lett. A 188 1994 263-271 hep-ph/9401117
20. J. Avan O. Babelon E. Billey The Gervais-Neveu-Felder equation and the quantum Calogero-Moser systems Commun. Math. Phys. 178 1996 281-299 hep-ph/9505091
21. J. Balog L. Dabrowski L. Fehér Classical r-matrix and exchange algebra in WZNW and Toda theories Phys. Lett. B 244 1990 227-234
22. G. Felder Conformal field theory and integrable systems associated to elliptic curves Proc. ICM Zürich 1994 Birkhäuser Basel 1247-1255 hep-ph/9407154
23. P. Etingof A. Varchenko Geometry and classification of solutions of the classical dynamical Yang-Baxter equation Commun. Math. Phys. 192 1998 77-120 q-alg/9703040.
24. L.-C. Li P. Xu Spin Calogero-Moser systems associated with simple Lie algebras C. R. Acad. Sci. Paris Ser. I 331 1 2000 55-61 math.SG/0009180;
25. L.-C. Li P. Xu Integrable spin Calogero-Moser systems Commun. Math. Phys. 231 2002 257-286 math.QA/0105162.
26. L.-C. Li A family of hyperbolic spin Calogero-Moser systems and the spin Toda lattices Commun. Pure Appl. Math. LVII 2004 791-832 math-ph/0506028
27. L.-C. Li Coboundary dynamical Poisson groupoids and integrable systems IMRN 51 2003 2725-2746 mathph/0506027.
28. L.-C. Li Poisson involutions, spin Calogero-Moser systems associated with symmetric Lie subalgebras and the symmetric space spin Ruijsenaars-Schneider model math-ph/0506025
29. L.-C. Li A class of integrable spin Calogero-Moser systems II: Exact solvability math-ph/0506026
30. A. Alekseev E. Meinrenken Clifford algebras and the classical dynamical Yang-Baxter equation Math. Res. Lett. 10 2003 253-268 math.RT/0209347..
31. N. Reshetikhin Degenerate integrability of spin Calogero-Moser systems and the duality with the spin Ruijsenaars systems Lett. Math. Phys. 63 2003 55-71 math.QA/0202245
32. D. Alekseevsky A. Kriegl M. Losik P.W. Michor The Riemannian geometry of orbit spaces. The metric, geodesics, and integrable systems Publ. Math. Debrecen 62 2003 247-276 math.DG/010215
33. J. Arnlind J. Hoppe Eigenvalue-dynamics off the Calogero-Moser system Lett. Math. Phys. 68 2004 121-129
34. S. Hochgerner Singular cotangent bundle reduction and spin Calogero-Moser systems math.SG/0411068
35. J. Blom E. Langmann Novel integrable spin-particle models from gauge theories on a cylinder Phys. Lett. B 429 1998 336-342 solv-int/9804007
36. J. Blom E. Langmann Finding and solving Calogero-Moser type systems using Yang-Mills gauge theories Nucl. Phys. B 563 1999 506-532 math-ph/ 9909019
37. A.P. Polychronakos Generalized Calogero-Sutherland systems from many-matrix models Nucl. Phys. B 546 1999 495-502 hep-th/9806189
38. P. Etingof O. Schiffmann On the moduli space of classical dynamical r-matrices Math. Res. Lett. 8 2001 157-170 math.QA/0005282
39. R. Sjamaar E. Lerman Stratified symplectic spaces and reduction Ann. Math. 134 1991 375-422
40. J.-P. Ortega T.S. Ratiu Momentum Maps and Hamiltonian Reduction 2004 Birkhäuser Basel
41. P. Libermann C.-M. Marle Symplectic Geometry and Analytical Mechanics 1986 Reidel Dordrecht
42. S. Mohrdieck R. Wendt Integral conjugacy classes of compact Lie groups Manuscripta Math. 114 2004 531-547 math.QA/0303118
43. V.G. Kac, Infinite-Dimensional Lie Algebras, third ed., Cambridge, 1990
44. G. Warner Harmonic Analysis on Semi-Simple Lie Groups, vol. I 1972 Springer Berlin
45. V.S. Gerdjikov A. Kyuldjiev G. Marmo G. Vilasi Real Hamiltonian forms of Hamiltonian systems nlin.SI/0310005
46. L. Fehér Dynamical r-matrices and Poisson-Lie symmetries in the chiral WZNW model Review presented at the Workshop on Integrable Theories, Solitons and Duality, IFT Sao Paulo, July 2002, JHEP Proceedings, PoS(unesp2002)012 hep-th/0212006
47. B. Enriquez P. Etingof Quantization of classical dynamical r-matrices with non-Abelian base Commun. Math. Phys. 254 2005 603-650 math.QA/ 0311224
48. L. Fehér A. Gábor B.G. Pusztai On dynamical r-matrices obtained from Dirac reduction and their generalizations to affine Lie algebras J. Phys. A 34 2001 7235-7248 math-ph/0105047
49. L. Fehér, B.G. Pusztai, Spin Calogero models and dynamical r-matrices, in: Proceedings of Symposium QTS-4, Varna, Bulgaria, August 2005, to appear, math-ph/0511059
50. Ya.S. Soibelman Irreducible representations of the function algebra on the quantum group SU(n), and Schubert cells Sov. Math. Dokl. 40 1 1990 34-38