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

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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

*Advances in Pure Mathematics*, Vol. 4 No. 1, 2014, pp. 1-10. doi: 10.4236/apm.2014.41001.

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