On the classification of Lorentzian holonomy groups

Abstract

If an (n + 2)-dimensional Lorentzian manifold is indecompos-able, but non-irreducible, then its holonomy algebra is contained in the parabolic algebra (R⊕so(n))⋉Rn. We show that its projec- tion onto so(n) is the holonomy algebra of a Riemannian manifold. This leads to a classification of Lorentzian holonomy groups and implies that the holonomy group of an indecomposable Lorentzian spin manifold with parallel spinor equals to G ⋉ Rn where G is a product of SU(p), Sp(q), G2 or Spin(7).