Contractions of Lie algebras and separation of variables. The n-dimensional sphere

Journal of Mathematical Physics

Volumn 40, Issue 3, 1999, Pages 1549-1573

  Izmest'ev, A A ^a   Pogosyan, G S ^a   Sissakian, A N ^a   Winternitz, P ^b  

^a JOINT INSTITUTE FOR NUCLEAR RESEARCH   (Russian Federation)

^b UNIVERSITÉ DE MONTRÉAL   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033241384     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532820     Document Type: Article

References (38)

1. A. A. Izmest'ev, G. S. Pogosyan, A. N. Sissakian, and P. Winternitz, "Contractions of Lie algebras and separation of variables," J. Phys. A 29, 5940-5962 (1996).
2. A. A. Izmest'ev, G. S. Pogosyan, A. N. Sissakian, and P. Winternitz, "Contractions of Lie algebras and separation of variables. Two-dimensional hyperboloid," Int. J. Mod. Phys. A 12, 53-61 (1997).
3. P. Winternitz and I. Fris, "Invariant expansions of relativistic amplitudes and subgroups of the proper Lorentz group," Sov. J. Nucl. Phys. 1, 636-643 (1965); Yad. Fiz. 1, 889-901 (1965).
4. P. Winternitz and I. Fris, "Invariant expansions of relativistic amplitudes and subgroups of the proper Lorentz group," Sov. J. Nucl. Phys. 1, 636-643 (1965); Yad. Fiz. 1, 889-901 (1965).
5. P. Winternitz, I. Lukac, and Ya. A. Smorodinskii, "Quantum numbers in the little groups of the Poincaré group," Sov. J. Nucl. Phys. 7, 139-145 (1968); Yad. Fiz. 7, 192-201 (1968).
6. P. Winternitz, I. Lukac, and Ya. A. Smorodinskii, "Quantum numbers in the little groups of the Poincaré group," Sov. J. Nucl. Phys. 7, 139-145 (1968); Yad. Fiz. 7, 192-201 (1968).
7. J. Patera and P. Winternitz, "A new basis for representations of the rotation group. Lamé and Heun polynomials," J. Math. Phys. 14, 1130-1139 (1973).
8. W. Miller, Jr., J. Patera, and P. Winternitz, "Subgroups of Lie groups and separation of variables," J. Math. Phys. 22, 251-260 (1991).
9. W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Reading, MA, 1997).
10. E. G. Kalnins, Separation of Variables for Riemannian Spaces of Constant Curvature (Longman, Burnt Mill, 1986).
11. N. Ya. Vilenkin, "Polyspherical and orispherical functions," Mat. Sb. 68, 432-443 (1965).
12. N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, RI, 1968).
13. N. Ya. Vilenkin, G. I. Kuznetsov, and Ya. A. Smorodinskii, "Eigenfunctions of the Laplace operator realizing representations of the groups U(2), SU(2), SO(3), U(3) and SU(3) and the symbolic method," Sov. J. Nucl. Phys. 2, 645-655 (1965); Yad. Fiz. 2, 906-917 (1965).
14. N. Ya. Vilenkin, G. I. Kuznetsov, and Ya. A. Smorodinskii, "Eigenfunctions of the Laplace operator realizing representations of the groups U(2), SU(2), SO(3), U(3) and SU(3) and the symbolic method," Sov. J. Nucl. Phys. 2, 645-655 (1965); Yad. Fiz. 2, 906-917 (1965).
15. M. S. Kildyushov, "Hyperspherical functions of the "tree" type in the n-particle problem," Sov. J. Nucl. Phys. 15, 113-123 (1972); Yad. Fiz. 15, 197-208 (1972).
16. M. S. Kildyushov, "Hyperspherical functions of the "tree" type in the n-particle problem," Sov. J. Nucl. Phys. 15, 113-123 (1972); Yad. Fiz. 15, 197-208 (1972).
17. G. I. Kuznetsov, S. S. Moskalyuk, Yu. F. Smirnov, and V. P. Shelest, A Graphical Theory of Representations of Orthogonal and Unitary Groups and its Physical Applications (Naukova Dumka, Kiev, 1992) (in Russian).
18. E. Inönü and E. P. Wigner, "On the contraction of groups and their representations," Proc. Natl. Acad. Sci. USA 39, 510-524 (1953).
19. E. J. Saletan, "Contraction of Lie groups," J. Math. Phys. 2, 1-21 (1961).
20. R. Gilmore, Lie Groups, Lie Algebras and Some of their Applications (Wiley, New York, 1974).
21. R. V. Moody and J. Patera, "Discrete and continuous graded contractions of representations of Lie algebras," J. Phys. A 24, 2227-2258 (1991).
22. X. Leng and J. Patera, "Graded contractions of representations of sl(n,C) with respect to the maximal parabolic subalgebras," J. Phys. A 27, 1233-1250 (1987).
23. M. Couture, J. Patera, R. T. Sharp, and P. Winternitz, "Graded contractions of sl(3,C)," J. Math. Phys. 32, 2310-2318 (1991).
24. M. Ait Abdelmalek, X. Leng, J. Patera, and P. Winternitz, "Grading refinements in the contractions of Lie algebras and their invariants," J. Phys. A 29, 7519-7543 (1996).
25. P. Winternitz, "Successive refinements of gradings and graded contractions of sl(3,C)," Int. J. Mod. Phys. A 12, 109-115 (1997).
26. J. Patera, G. Pogosyan, and P. Winternitz, "Graded contractions of the Lie algebra e(2,1)," CRM-2484, 1997.
27. E. Weimar-Woods, "Contractions of Lie algebras: generalized Inönü-Wigner contractions versus graded contractions," J. Math. Phys. 36, 4519-4548 (1995).
28. L. P. Eisenhart, "Separable systems of Stäckel," Ann. Math. 35, 284-305 (1934).
29. D. A. Suprunenko and R. I. Tyshkevich, Commutative Matrices (Academic, New York, 1968).
30. J. Patera, P. Winternitz, and H. Zassenhaus, "Maximal Abelian subalgebras of real and complex symplectic Lie algebras," J. Math. Phys. 24, 1973-1985 (1983).
31. M. A. del Olmo, M. A. Rodriguez, P. Winternitz, and H. Zassenhaus, "Maximal Abelian subalgebras of pseudounitary Lie algebras," Linear Algebr. Appl. 135, 79-151 (1990).
32. V. Hussin, P. Winternitz, and H. Zassenhaus, "Maximal Abelian subalgebras of complex orthogonal Lie algebras," Linear Algebr. Appl. 141, 183-220 (1990).
33. V. Hussin, P. Winternitz, and H. Zassenhaus, "Maximal Abelian subalgebras of pseudoorthogonal Lie algebras," Linear Algebr. Appl. 173, 125-163 (1992).
34. E. G. Kalnins and P. Winternitz, "Maximal Abelian subalgebras of complex Euclidean Lie algebras," Can. J. Phys. 72, 389-404 (1994).
35. Z. Thomova and P. Winternitz, "Maximal Abelian subgroups of the isometry and conformal groups of Euclidean and Minkowski spaces," J. Phys. A 31, 1831-1859 (1998).
36. 2u + λu = 0 allows a complete separation of variables," Mat. Sb. 27, 379-426 (1950) (in Russian).
37. E. Kalnins, W. Miller, Jr., and P. Winternitz, "The group O(4), separation of variables and the hydrogen atom," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 30, 630-664 (1976).
38. G. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw.-Hill, New York, 1953).