TITLE:
Contractions of Certain Lie Algebras in the Context of the DLF-Theory
AUTHORS:
Alexander Levichev, Oleg Sviderskiy
KEYWORDS:
Lie Algebras with Invariant Lorentzian Forms; Lorentzian Symmetric Spaces; Contractions of Lie Algebras; Conformal Lie Algebra; Segal’s Chronometric Theory; DLF-Theory
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.4 No.1,
January
8,
2014
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).