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 $N_1\left(k\right)=R{⋉}_{F_1}{h}_{2k+1}$ and $N_2\left(k\right)=R{⋉}_{F_2}\left(h_{2k+1}\oplus R\right)$, defined by some particular derivations $F_1,F_2$. In all those cases we also find the equivalence classes of flat Lorentzian products.