How to find the holonomy algebra of a Lorentzian manifold

TL;DR

This paper presents a method to determine the holonomy algebra of Lorentzian manifolds, reducing the problem to indecomposable cases and classifying possible algebras based on geometric structures.

Contribution

It introduces a systematic approach and criteria to identify the holonomy algebra of Lorentzian manifolds, including an algorithm to find associated subalgebras.

Findings

Classification of four types of holonomy algebras for Lorentzian manifolds

Criteria for identifying the type of holonomy algebra

Algorithm for determining the associated subalgebra 

Abstract

Manifolds with exceptional holonomy play an important role in string theory, supergravity and M-theory. It is explained how one can find the holonomy algebra of an arbitrary Riemannian or Lorentzian manifold. Using the de~Rham and Wu decompositions, this problem is reduced to the case of locally indecomposable manifolds. In the case of locally indecomposable Riemannian manifolds, it is known that the holonomy algebra can be found from the analysis of special geometric structures on the manifold. If the holonomy algebra $\mathfrak{g}\subset\mathfrak{so}(1,n-1)$ of a locally indecomposable Lorentzian manifold $(M,g)$ of dimension $n$ is different from $\mathfrak{so}(1,n-1)$, then it is contained in the similitude algebra $\mathfrak{sim}(n-2)$. There are 4 types of such holonomy algebras. Criterion how to find the type of $\mathfrak{g}$ are given, and special geometric structures corresponding to each type are described. To each $\mathfrak{g}$ there is a canonically associated subalgebra $\mathfrak{h}\subset\mathfrak{so}(n-2)$. An algorithm how to find $\mathfrak{h}$ is provided.