Holonomy algebras of Einstein pseudo-Riemannian manifolds

Abstract

The holonomy algebras of Einstein not Ricci-flat pseudo-Riemannian manifolds of arbitrary signature are classified. As illustrating examples, the cases of Lorentzian manifolds, pseudo-Riemannian manifolds of signature $(2,n)$ and the para-quaternionic-K\"ahlerian manifolds with non-zero scalar curvature are considered. Einstein not Ricci-flat metrics of signature $(2,n)$ with all possible holonomy algebras are given.