About the classification of the holonomy algebras of Lorentzian manifolds

TL;DR

This paper discusses the classification of holonomy algebras of Lorentzian manifolds, showing that for certain cases, the list matches that of Riemannian manifolds, and provides a direct proof for semisimple non-simple cases.

Contribution

It offers a simple direct proof that the classification of certain Lorentzian holonomy algebras coincides with Riemannian cases for semisimple non-simple Lie algebras.

Findings

Classification reduces to irreducible subalgebras satisfying a Bianchi-like identity

Leistner's list matches Riemannian holonomy algebras

Provides a direct proof for semisimple non-simple Lie algebras

Abstract

The classification of the holonomy algebras of Lorentzian manifolds can be reduced to the classification of irreducible subalgebras $\mathfrak{h}\subset\mathfrak{so}(n)$ that are spanned by the images of linear maps from $\mathbb{R}^n$ to $\mathfrak{h}$ satisfying an identity similar to the Bianchi one. T. Leistner found all such subalgebras and it turned out that the obtained list coincides with the list of irreducible holonomy algebras of Riemannian manifolds. The natural problem is to give a simple direct proof to this fact. We give such proof for the case of semisimple not simple Lie algebras $\mathfrak{h}$.