Hidden algebras of the (super) Calogero and Sutherland models

Journal of Mathematical Physics

Volumn 39, Issue 3, 1998, Pages 1285-1315

  Brink, Lars ^a   Turbiner, Alexander ^b,c   Wyllard, Niclas ^a  

^a CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

^b UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO   (Mexico)

^c INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032351485     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532380     Document Type: Article

References (39)

1. F. Calogero, "Solution of a three-body problem in one dimension," J. Math. Phys. 10, 2191-2196 (1996); "Ground state of a one-dimensional N-body problem," ibid. 10, 2197-2200 (1969); "Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials," ibid. 12, 419-436 (1971).
2. F. Calogero, "Solution of a three-body problem in one dimension," J. Math. Phys. 10, 2191-2196 (1996); "Ground state of a one-dimensional N-body problem," ibid. 10, 2197-2200 (1969); "Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials," ibid. 12, 419-436 (1971).
3. F. Calogero, "Solution of a three-body problem in one dimension," J. Math. Phys. 10, 2191-2196 (1996); "Ground state of a one-dimensional N-body problem," ibid. 10, 2197-2200 (1969); "Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials," ibid. 12, 419-436 (1971).
4. B. Sutherland, "Exact results for a quantum many-body problem in one dimension," Phys. Rev. A 4, 2019-2021 (1971); "Exact results for a quantum many-body problem in one dimension, II," ibid. 5, 1372-1376 (1972).
5. B. Sutherland, "Exact results for a quantum many-body problem in one dimension," Phys. Rev. A 4, 2019-2021 (1971); "Exact results for a quantum many-body problem in one dimension, II," ibid. 5, 1372-1376 (1972).
6. D. Z. Freedman and P. F. Mende, "An exactly solvable N particle system in supersymmetric quantum mechanics," Nucl. Phys. B 344, 317-343 (1990).
7. L. Brink, T. H. Hansson, S. Konstein, and M. A. Vasiliev, "The Calogero model: Anyonic representation, fermionic extension and supersymmetry," Nucl. Phys. B 401, 591-612 (1993), .
8. 2 system," Phys. Rev. Lett. 70, 4029-4033 (1993), cond-mat/9212029.
9. M. A. Olshanetsky and A. M. Perelomov, "Quantum completely integrable systems connected with semi-simple Lie algebras," Lett. Math. Phys. 2, 7-13 (1977).
10. D. Kazhdan, B. Kostant, and S. Sternberg, "Hamiltonian group actions and dynamical systems of Calogero type," Commun. Pure Appl. Math. 31, 481-507 (1978).
11. M. A. Olshanetsky and A. M. Perelomov, "Classical integrable finite dimensional systems related to Lie algebras," Phys. Rep. 71, 313 (1981).
12. M. A. Olshanetsky and A. M. Perelomov, "Quantum integrable systems related to Lie algebras," Phys. Rep. 94, 313 (1983).
13. A. Gorsky and N. Nekrasov, "Hamiltonian systems of Calogero type and two-dimensional Yang-Mills theory," Nucl. Phys. B 414, 213-238 (1994), hep-th/9304047; "Relativistic Calogero-Moser model as gauged WZW theory," ibid. 436, 582-608 (1995), hep-th/9401017.
14. A. Gorsky and N. Nekrasov, "Hamiltonian systems of Calogero type and two-dimensional Yang-Mills theory," Nucl. Phys. B 414, 213-238 (1994), hep-th/9304047; "Relativistic Calogero-Moser model as gauged WZW theory," ibid. 436, 582-608 (1995), hep-th/9401017.
15. A. V. Turbiner, "Lie algebras and linear operators with invariant subspace," in Lie Algebras, Cohomologies and New Findings in Quantum Mechanics, edited by N. Kamran and P. J. Olver (AMS, 1994), Vol. 160, pp. 263-310; "Lie-algebras and Quasi-exactly-solvable Differential Equations," in CRC Handbook of Lie Group Analysis of Differential Equations, edited by N. Ibragimov (CRC, Boca Raton, FL, 1995), Vol. 3, Chap. 12, pp. 331-366, 1995; hep-th/9409068.
16. A. V. Turbiner, "Lie algebras and linear operators with invariant subspace," in Lie Algebras, Cohomologies and New Findings in Quantum Mechanics, edited by N. Kamran and P. J. Olver (AMS, 1994), Vol. 160, pp. 263-310; "Lie-algebras and Quasi-exactly-solvable Differential Equations," in CRC Handbook of Lie Group Analysis of Differential Equations, edited by N. Ibragimov (CRC, Boca Raton, FL, 1995), Vol. 3, Chap. 12, pp. 331-366, 1995; hep-th/9409068.
17. W. Rühl and A. Turbiner, "Exact solvability of the Calogero and Sutherland models," Mod. Phys. Lett. A 10, 2213-2222 (1995), hep-th/9506105.
18. L. Brink, T. H. Hansson, and M. A. Vasiliev, "Explicit solution to the N body Calogero problem," Phys. Lett. B 286, 109-111 (1992), hep-th/9206049.
19. L. Lapointe and L. Vinet, "Exact operator solutions of the Calogero-Sutherland model," Commun. Math. Phys. 178, 425-452 (1996), .
20. D. Bernard, V. Pasquier, and D. Serban, "Exact solution of long-range interacting spin chains with boundaries," Europhys. Lett. 30, 301-305 (1995), hep-th/9501044.
21. D. Serban, "Some properties of Calogero-Sutherland model with reflections," J. Phys. A 30, 4215-4226 (1997), hep-th/9701118.
22. H. Awata, Y. Matsuo, S. Odake, and J. Shiraishi, "Collective field theory, Calogero-Sutherland model and generalized matrix models," Phys. Lett. B 347, 49-55 (1995), hep-th/9411053; "Excited states of Calogero-Sutherland model and singular vectors of the W(N) algebra," Nucl. Phys. B 449, 347-374 (1995), hep-th/9503043; K. Mimachi and Y. Yamada, "Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials," Commun. Math. Phys. 174, 447-455 (1995).
23. H. Awata, Y. Matsuo, S. Odake, and J. Shiraishi, "Collective field theory, Calogero-Sutherland model and generalized matrix models," Phys. Lett. B 347, 49-55 (1995), hep-th/9411053; "Excited states of Calogero-Sutherland model and singular vectors of the W(N) algebra," Nucl. Phys. B 449, 347-374 (1995), hep-th/9503043; K. Mimachi and Y. Yamada, "Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials," Commun. Math. Phys. 174, 447-455 (1995).
24. H. Awata, Y. Matsuo, S. Odake, and J. Shiraishi, "Collective field theory, Calogero-Sutherland model and generalized matrix models," Phys. Lett. B 347, 49-55 (1995), hep-th/9411053; "Excited states of Calogero-Sutherland model and singular vectors of the W(N) algebra," Nucl. Phys. B 449, 347-374 (1995), hep-th/9503043; K. Mimachi and Y. Yamada, "Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials," Commun. Math. Phys. 174, 447-455 (1995).
25. H. Jack, "A class of symmetric polynomials with aparameter," Proc. R. Soc. Edinburgh Sect. A 69, 1-17 (1969-1970).
26. R. Stanley, "Some combinatorial properties of Jack symmetric functions," Adv. Math. 77, 76-115 (1989).
27. I. G. MacDonald, Symmetric Functions and Hall Polynomials, 2nd ed. (Clarendon, Oxford, 1995).
28. A. M. Perelomov, "Algebraic approach to the solution of a one-dimensional model of N interacting particles," Teor. Mat. Fiz. 6, 364 (1971) (in Russian); English translation: Sov. Phys. - Theor Math. Phys. 6, 263 (1971).
29. A. M. Perelomov, "Algebraic approach to the solution of a one-dimensional model of N interacting particles," Teor. Mat. Fiz. 6, 364 (1971) (in Russian); English translation: Sov. Phys. - Theor Math. Phys. 6, 263 (1971).
30. J. Wolfes, "On the three-body linear problem with three-body interaction," J. Math. Phys. 15, 1420-1424 (1974).
31. C. Quesne, "An exactly solvable three particle problem with three body interaction," Phys. Rev. A 55, 3391-3394 (1997).
32. A. Capella, M. Rosenbaum, and A. Turbiner, "Solvability of G(2) integrable system" preprint ICN-UNAM 97/05 1997, 16, (to appear in Int. J. Mod. Phys. A).
33. J. A. Minahan and A. P. Polychronakos, "Integrable systems for particles with internal degrees of freedom," Phys. Lett. B 302, 265-270 (1993).
34. A. Turbiner, "Quasi-Exactly-Solvable Problems and the SL(2,R) Group, preprint ITEP-197 (1987); Commun. Math. Phys. 118, 467-474 (1988).
35. A. Turbiner, "Quasi-Exactly-Solvable Problems and the SL(2,R) Group, preprint ITEP-197 (1987); Commun. Math. Phys. 118, 467-474 (1988).
36. A. Morozov, A. Perelomov, A. Rosly, M. Shifman, and A. Turbiner, "Quasi-Exactly-Solvable Problems: One-Dimensional Analogue of Rational Conformal Field Theories," Int. J. Mod. Phys. A 5, 803-843 (1990).
37. A. Minzoni, M. Rosenbaum, and A. Turbiner, "Quasi-Exactly-Solvable Many-Body Problems," Mod. Phys. Lett. A 11, 1977-1984 (1996), hep-th/9606092.
38. A. Turbiner, "Lame Equation and sl(2)," preprint ITEP-109 (1988); "Lame Equation, sl(2) and Isospectral Deformation," J. Phys. A 22, L1-L3 (1989).
39. A. Turbiner, "Lame Equation and sl(2)," preprint ITEP-109 (1988); "Lame Equation, sl(2) and Isospectral Deformation," J. Phys. A 22, L1-L3 (1989).