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

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).

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.

A. Levichev and O. Sviderskiy, "Contractions of Certain Lie Algebras in the Context of the DLF-Theory,"

References

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[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.

[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.