Share This Article:

Contractions of Certain Lie Algebras in the Context of the DLF-Theory

DOI: 10.4236/apm.2014.41001    3,254 Downloads   5,261 Views  


Contractions of the Lie algebras d = u(2), f = u(1 ,1) to the oscillator Lie algebra l are realized via the adjoint action of SU(2,2) when d, l, f are viewed as subalgebras of su(2,2). Here D, L, F are the corresponding (four-dimensional) real Lie groups endowed with bi-invariant metrics of Lorentzian signature. Similar contractions of (seven-dimensional) isometry Lie algebras iso(D), iso(F) to iso(L) are determined. The group SU(2,2) acts on each of the D, L, F by conformal transformation which is a core feature of the DLF-theory. Also, d and f are contracted to T, S-abelian subalgebras, generating parallel translations, T, and proper conformal transformations, S (from the decomposition of su(2,2) as a graded algebra T + Ω + S, where Ω is the extended Lorentz Lie algebra of dimension 7).

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

A. Levichev and O. Sviderskiy, "Contractions of Certain Lie Algebras in the Context of the DLF-Theory," Advances in Pure Mathematics, Vol. 4 No. 1, 2014, pp. 1-10. doi: 10.4236/apm.2014.41001.


[1] A. K. Guts and A. V. Levichev, “On the Foundations of Relativity Theory,” Doklady Akademii Nauk SSSR, Vol. 277, No. 6, 1984, pp. 1299-1303. (in Russian)
[2] A. V. Levichev, “Causal Cones in Low-Dimensional Lie Algebras,” Siberian Journal of Mathematics, Vol. 26, No. 5, 1985, pp. 192-195. (in Russian)
[3] S. Paneitz and I. Segal, “Analysis in Space-Time Bundles I: General Considerations and the Scalar Bundle,” Journal of Functional Analysis, Vol. 47, No. 1, 1982, pp. 78-142.
[4] A. V. Levichev, “Certain Symmetric General Relativistic Space-Times as the Solutions to the Einstein-Yang-Mills Equations,”, Proceedings Group Theoretical Methods in Physics (III International Seminar), Yurmala, 1985, pp. 145-150. (in Russian)
[5] D. Kramer, H. Stephani, M. MacCallum and E. Herlt, “Exact Solutions of Einstein’s Field Equations,” VEB Deutscher Verlag der Wissenschaften, Berlin, 1980.
[6] A. V. Levichev, “Chronogeometry of an Electromagnetic Wave Defined by a Bi-Invariant Metric on the Oscillator Lie Group,” Siberian Journal of Mathematics, Vol. 27, No. 2, 1986, pp. 237-245.
[7] J. Hilgert, K. H. Hofmann and J. D. Lawson, “Lie Groups, Convex Cones, and Semigroups,” Clarendon Press, Oxford, 1989.
[8] A. V. Levichev, “Three Symmetric Worlds Instead of the Minkowski Space-Time,” Transactions on RANS, series MMM&C, Vol. 7, No. 3-4, 2003, pp. 87-93.
[9] A. V. Levichev, “Pseudo-Hermitian Realization of the Minkowski World through the DLF-Theory,” Physica Scripta, Vol. 83, No. 1, 2011, pp. 1-9.
[10] V. Guillemin and S. Sternberg, “Geometric Asymptotics,” American Mathematical Society, Providence, 1977.
[11] A. Fialowski and M. De Montigny, “On Deformations and Contractions of Lie Algebras,” SIGMA, Vol. 2, 2006, p. 10.
[12] I. Segal, “A Class of Operator Algebras Which Are Determined by Groups,” Duke Mathematical Journal, Vol. 18, No. 1, 1951, pp. 221-265.
[13] A. Knapp, “Representation Theory of Semisimple Groups: An Overview Based on Examples,” Princeton University Press, Princeton, 2001.
[14] I. E. Segal, H. P. Jakobsen, B. Orsted, S. M. Paneitz and B. Speh, “Covariant Chronogeometry and Extreme Distances: Elementary Particles,” Proceedings of the National Academy of Sciences, Vol. 78, No. 9, 1981, pp. 5261-5265.
[15] S. Sternberg, “Chronogeometry and Symplectic Geometry,” Colloques Internationaux C.N.R.S. Geometrie Symplectique et Physique Mathematique, Vol. 237, 1975, pp. 45-57.
[16] M. Cahen, and N. Wallach, “Lorentzian Symmetric Spaces,” Bulletin of the American Mathematical Society, Vol. 76, No. 3, 1970, pp. 585-591.
[17] R. F. Streater, “The Representations of the Oscillator Group,” Communications in Mathematical Physics, Vol. 4, No. 3, 1967, pp. 217-236.
[18] A. Medina and Ph. Revoy, “Les Groups Oscillateurs at Leurs Reseaux,” Manuscripta Mathematica, Vol. 52, No. 1-3, 1985, pp. 81-95.
[19] M. Cahen and Y. Kerbrat, “Champs des Vecteurs Conformes et Transformations Conformes des Espace Lorentziens Symmetriques,” Journal de Mathématiques Pures et Appliquées, Vol. 4, No. 57, 1978, pp. 99-132.

comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.