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

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

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).

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.

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