Journal of Lie Theory 22 (2012), No. 1, 269--289
Copyright Heldermann Verlag 2012
Left-invariant Lorentzian Flat Metrics on Lie Groups
Malika Ait Ben Haddou
Faculté des Sciences, Meknes, Morocco
maitbenha@gmail.com
Mohamed Boucetta
Université Cadi-Ayyad, Faculté des Sciences Gueliz, Marrakech, Morocco
mboucetta2@yahoo.fr
Hicham Lebzioui
Faculté des Sciences, Meknes, Morocco
hlebzioui@gmail.com
We call the Lie algebra of a Lie group with a left invariant pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new proof of a classical result of Milnor on Riemannian flat Lie algebras. We reduce the study of Lorentzian flat Lie algebras to those with trivial center or those with degenerate center. We show that the double extension process can be used to construct all Lorentzian flat Lie algebras with degenerate center generalizing a result of Aubert-Medina on Lorentzian flat nilpotent Lie algebras. Finally, we give the list of Lorentzian flat Lie algebras with degenerate center up to dimension 6.
Keywords: Coadjoint orbit, integral coadjoint orbit.
MSC: 53C50, 16T25; 53C20, 17B62
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