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 indecom...

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Veröffentlicht in:The Bulletin of the London Mathematical Society 2024-06, Vol.56 (6), p.2132-2149
Hauptverfasser: Bajo, Ignacio, Benayadi, Saïd, Lebzioui, Hicham
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Sprache:eng
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Zusammenfassung: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 h2k+1${\mathfrak {h}}_{2k+1}$ denotes the 2k+1$2k+1$‐dimensional Heisenberg Lie algebra, then the only non‐abelian Lie algebras admitting flat Lorentzian metrics which are indecomposable are h3${\mathfrak {h}}_3$ and the semidirect products N1(k)=R⋉F1h2k+1${\mathfrak {N}}_1(k)={\mathbb {R}}\ltimes _{ F_1}{\mathfrak {h}}_{2k+1}$ and N2(k)=R⋉F2(h2k+1⊕R)${\mathfrak {N}}_2(k)={\mathbb {R}}\ltimes _{ F_2}({\mathfrak {h}}_{2k+1}\oplus {\mathbb {R}})$, defined by some particular derivations F1,F2$F_1,F_2$. In all those cases we also find the equivalence classes of flat Lorentzian products.
ISSN:0024-6093
1469-2120
DOI:10.1112/blms.13047