The holonomy algebra $\g$ of an indecomposable Lorentzian (n+2)-dimensional
manifold $M$ is a weakly-irreducible subalgebra of the Lorentzian algebra
$\so_{1,n+1}$.
L. Berard Bergery and A. Ikemakhen divided weakly-irreducible not irreducible
subalgebras into 4 types and associated with each such subalgebra $\g$ a
subalgebra $\h\subset \so_n$ of the orthogonal Lie algebra. We give a
description of the spaces $\R(\g)$ of the curvature tensors for algebras of
each type in terms of the space $\P(\h)$ of $\h$-valued 1-forms on $\Real^n$
that satisfy the Bianchi identity and reduce the classification of the holonomy
algebras of Lorentzian manifolds to the classification of irreducible
subalgebras $\h$ of $so(n)$ with $L(\P(\h))=\h$. We prove that for $n\leq 9$
any such subalgebra $\h$ is the holonomy algebra of a Riemannian manifold. This
gives a classification of the holonomy algebras for Lorentzian manifolds $M$ of
dimension $\leq 11$.
@article{0304407,
author = {Galaev, Anton S.},
title = {The spaces of curvature tensors for holonomy algebras of Lorentzian
manifolds},
journal = {arXiv},
volume = {2003},
number = {0},
year = {2003},
language = {en},
url = {http://dml.mathdoc.fr/item/0304407}
}
Galaev, Anton S. The spaces of curvature tensors for holonomy algebras of Lorentzian
manifolds. arXiv, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/0304407/