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

Sigbjørn Hervik

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.

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

Abstract

We continue to study left-invariant pseudo-Riemannian metrics on Lie groups being in the null cone of the -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 and we find that all solvable Lie algebras, and non-trivially Levi-decomposable Lie algebras, of dimension are in the null cone, \emph{except} the 3-dimensional solvable Lie algebra . For semi-simple, we also give a construction where is in the null cone and give examples of such spaces for \emph{all} the real simple Lie algebras . For example, for the exceptional split groups this construction places the split , split and split in the null cone of the , and action, respectively, and hence, their corresponding left-invariant pseudo-Riemannian metrics are VSI.

Paper Structure

This paper contains 17 sections, 2 theorems, 35 equations, 4 tables.

Table of Contents

  1. Introduction
  2. The moving bracket approach
  3. The null cone
  4. The boost weight decomposition and the null cone
  5. Lie algebras of dimension $\leq 3$
  6. Lie algebras of dimension 4
  7. Lie algebras of dimension 5
  8. Lie algebras of dimension 6
  9. Solvable Lie algebras of dimension 6
  10. Lie algebras ${\mathfrak s}_{2,1}\oplus{\mathfrak s}_{2,1}\oplus{\mathfrak s}_{2,1}$, ${\mathfrak s}_{4,\bullet}\oplus{\mathfrak s}_{2,1}$:
  11. Lie algebras ${\mathfrak s}_{3,\bullet}\oplus{\mathfrak s}_{3,\bullet}$:
  12. Lie algebras ${\mathfrak s}_{6,\bullet}$:
  13. Direct sum decomposable Lie algebras of dimension 6
  14. Non-trivial Levi decomposable Lie algebras of dimension 6
  15. Semi-simple 6-dimensional Lie algebras
  16. ...and 2 more sections

Key Result

Theorem 1.1

The following classes of Lie algebras are in the null cone of some $O(p,q)$-action:

Theorems & Definitions (2)

  • Theorem 1.1
  • Theorem 2.1