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

Alexander Levichev, Oleg Sviderskiy

Sobolev Institute of Mathematics, Novosibirsk, Russia.

DOI: 10.4236/apm.2014.41001                

Abstract

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).

Keywords

Lie Algebras with Invariant Lorentzian Forms; Lorentzian Symmetric Spaces; Contractions of Lie Algebras; Conformal Lie Algebra; Segal’s Chronometric Theory; DLF-Theory

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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. http://dx.doi.org/10.1016/0022-1236(82)90101-X
[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. http://dx.doi.org/10.1007/BF00969391
[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. http://dx.doi.org/10.1090/surv/014
[11]	A. Fialowski and M. De Montigny, “On Deformations and Contractions of Lie Algebras,” SIGMA, Vol. 2, 2006, p. 10. http://www.emis.de/journals/SIGMA/2006/Paper048/
[12]	I. Segal, “A Class of Operator Algebras Which Are Determined by Groups,” Duke Mathematical Journal, Vol. 18, No. 1, 1951, pp. 221-265. http://dx.doi.org/10.1215/S0012-7094-51-01817-0
[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. http://dx.doi.org/10.1073/pnas.78.9.5261
[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. http://dx.doi.org/10.1090/S0002-9904-1970-12448-X
[17]	R. F. Streater, “The Representations of the Oscillator Group,” Communications in Mathematical Physics, Vol. 4, No. 3, 1967, pp. 217-236. http://dx.doi.org/10.1007/BF01645431
[18]	A. Medina and Ph. Revoy, “Les Groups Oscillateurs at Leurs Reseaux,” Manuscripta Mathematica, Vol. 52, No. 1-3, 1985, pp. 81-95. http://dx.doi.org/10.1007/BF01171487
[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.