Left-invariant Pseudo-Riemannian metrics on Lie groups: The null cone II

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Abstract

We continue to study left-invariant pseudo-Riemannian metrics on Lie groups being in the null cone of the $O(p,q)$-action using the moving bracket approach. In particular, the Lie algebra being in the null cone implies that the pseudo-Riemannian metric have all vanishing scalar curvature invariants (VSI). We consider \emph{all} Lie algebras of dimension $\leq 6$ and we find that all solvable Lie algebras, and non-trivially Levi-decomposable Lie algebras, of dimension $\leq 6$ are in the null cone, \emph{except} the 3-dimensional solvable Lie algebra $\mathfrak{s}_{3,3}$. For $\mathfrak{g}$ semi-simple, we also give a construction where $\mathfrak{g}\oplus\mathbb{R}^m$ is in the null cone and give examples of such spaces for \emph{all} the real simple Lie algebras $\mathfrak{g}$. For example, for the exceptional split groups this construction places the split $\mathfrak{e}_6\oplus\mathbb{R}^6$, split $\mathfrak{e}_7\oplus\mathbb{R}^7$ and split $\mathfrak{e}_8\oplus\mathbb{R}^8$ in the null cone of the $O(42,42)$, $O(70.70)$ and $O(128,128)$ action, respectively, and hence, their corresponding left-invariant pseudo-Riemannian metrics are VSI.