The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds

The holonomy algebra g of an indecomposable Lorentzian (n+2)-dimensional manifold M is a weakly-irreducible subalgebra of the Lorentzian algebra so1,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⊂son 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 Rn 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⩽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 ⩽11.