Classification of flat Lorentzian nilpotent Lie algebras

We give a complete classification of flat Lorentzian nilpotent Lie algebras, this is to say of pseudo-Euclidean Lie algebras associated to nilpotent Lie groups endowed with a left-invariant Lorentzian metric of vanishing curvature. We prove that every such a Lie algebra is a direct sum of an indecomposable flat Lorentzian Lie algebra and an abelian Euclidean summand and show that, if h(2k+1) denotes the 2k+1-dimensional Heisenberg Lie algebra, then the only non-abelian Lie algebras admitting flat Lorentzian metrics which are indecomposable are h(3) and the semidirect products n(1)(k) = R(sic)(F1) h(2k+1) and n(2)(k)= R(sic)(F2) (h(2k+1) circle plus R) F-1,F-2. In all those cases we also find the equivalence classes of flat Lorentzian products.