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


SIGMA 2 (2006), 028, 11 pages      math-ph/0602065      http://dx.doi.org/10.3842/SIGMA.2006.028

Application of the Gel'fand Matrix Method to the Missing Label Problem in Classical Kinematical Lie Algebras

Rutwig Campoamor-Stursberg
Departamento Geometría y Topología, Fac. CC. Matemáticas U.C.M., Plaza de Ciencias 3, E-28040 Madrid, Spain

Received November 06, 2005, in final form February 14, 2006; Published online February 28, 2006

Abstract
We briefly review a matrix based method to compute the Casimir operators of Lie algebras, mainly certain type of contractions of simple Lie algebras. The versatility of the method is illustrated by constructing matrices whose characteristic polynomials provide the invariants of the kinematical algebras in (3+1)-dimensions. Moreover it is shown, also for kinematical algebras, how some reductions on these matrices are useful for determining the missing operators in the missing label problem (MLP).

Key words: Casimir operator; characteristic polynomial; Lie algebra; missing label; kinematical group.

pdf (231 kb)   ps (176 kb)   tex (14 kb)

References

  1. Gel'fand I.M., The center of an infinitesimal group ring, Mat. Sbornik, 1950, V.26, 103-112.
  2. Perelomov A.M., Popov V.S., Casimir operators for semisimple Lie groups, Izv. Akad. Nauk SSSR Ser. Mat., 1968, V.32, 1368-1390.
  3. Barannik L.F., Fushchich W.I., Casimir operators for the generalized Poincaré and Galilei groups, in Proceedings of the Third International Seminar "Group Theoretical Methods in Physics" (May 22-24, 1985, Yurmala), Moscow, Nauka, 1986, 176-183.
  4. Patera J., Sharp R.T., Winternitz P., Zassenhaus H., Continuous subgroups of physics III. The de Sitter groups, J. Math. Phys., 1977, V.18, 2259-2288.
  5. Campoamor-Stursberg R., A new matrix method for the Casimir operators of the Lie algebras wsp(N,R) and Isp(2N,R), J. Phys. A: Math. Gen., 2005, V.38, 4187-4208.
  6. Campoamor-Stursberg R., Über die Struktur der Darstellungen komplexer halbeinfacher Lie-Algebren, die mit einer Heisenberg-Algebra verträglich sind, Acta Phys. Polon. B, 2005, V.36, 2869-2886.
  7. Quesne C., Casimir operators of semidirect sum Lie algebras, J. Phys. A: Math. Gen., 1988, V.21, L321-L324.
  8. Campoamor-Stursberg R., Intrinsic formulae for the Casimir operators of semidirect products of the exceptional Lie algebra G2 and a Heisenberg Lie algebra, J. Phys. A: Math. Gen., 2004, V.37, 9451-9466.
  9. Iwahori N., On real irreducible representations of Lie algebras, Nagoya Math. J., 1959, V.14, 59-83.
  10. Bacry H., Lévy-Leblond J.M., Possible kinematics, J. Math. Phys., 1968, V.9, 1605-1614.
  11. Lôhmus J., Tammelo R., Contractions and deformations of space-time algebras I, Hadronic J., 1997, V.20, 361-416.
  12. Gromov N.A., Contractions and analytic continuations of the classical groups. A unified approach, Syktyvkar, Akad. Nauk SSSR Ural. Otdel., 1990 (in Russian).
  13. Herranz F.J., Santander M., Casimir invariants for the complete family of quasisimple orthogonal Lie algebras, J. Phys. A: Math. Gen, 1997, V.30, 5411-5426, physics/9702032.
  14. Sharp R.T., Internal-labelling operators, J. Math. Phys., 1975, V.16, 2050-2053.
  15. Peccia A., Sharp R.T., Number of independent missing label operators, J. Math. Phys., 1976, V.17, 1313-1314.
  16. Campoamor-Stursberg R., The structure of the invariants of perfect Lie algebras, J. Phys. A: Math. Gen., 2003, V.36, 6709-6723.
  17. Winternitz P., Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Contractions of Lie algebras and separation of variables, Fiz. Elementar. Chastits i Atom. Yadra, 2001, V.32, 84-87.

Previous article   Next article   Contents of Volume 2 (2006)