Contractions of Certain Lie Algebras in the Context of the DLF-Theory
- 1 Sobolev Institute of Mathematics, Novosibirsk, Russia
- 2 Sobolev Institute of Mathematics, Novosibirsk, Russia
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).
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