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).
Highlights
IntroductionAs noticed by the first author (see [1,2]), there are precisely three four-dimensional non-abelian Lie algebras that admit a non-degenerate invariant bilinear form of Lorentzian signature: the oscillator Lie algebra l, defined by the following commutation relations in a certain basis l1, l2, l3, l4:
This DLF-theory can be briefly characterized as the LF-modification of Segal’s Chronometry
V4 = – (1 2) ( L−10 − L34 − L12 ). This subalgebra is isomorphic to the oscillator Lie algebra since the commutation relations are the same as (1.1)
Summary
As noticed by the first author (see [1,2]), there are precisely three four-dimensional non-abelian Lie algebras that admit a non-degenerate invariant bilinear form of Lorentzian signature: the oscillator Lie algebra l, defined by the following commutation relations in a certain basis l1, l2, l3, l4:. Certain Lie groups (corresponding to the Lie algebras d, l, f), endowed with bi-invariant metrics of Lorentzian signature, provide so-called homogeneous solutions to Einstein’s equations of General Relativity Theory, GRT. We denote these solutions by the corresponding capital letters. This DLF-theory can be briefly characterized as the LF-modification of Segal’s Chronometry (the latter one is based on D, see [3,9] and references therein) It is known (see [9]) that these three space-times are conformally flat, and that in each case the isometry group of the corresponding Lorentzian manifold is of dimension 7. Regarding contractions of Lie algebras, we follow [11]
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