Holonomy algebras of pseudo-hyper-Kählerian manifolds of index 4

Abstract

The holonomy algebra of a pseudo-hyper-Kählerian manifold of signature (4, 4n + 4) is a subalgebra of sp(1, n + 1). Possible holonomy algebras of these manifolds are classified. Using this, a new proof of the classification of simply connected pseudo-hyper-Kählerian symmetric spaces of index 4 is obtained.

Holonomy groups of Lorentzian manifolds

“…В случае псевдоримановых многообразий сигнатур, отличных от римановой и лоренцевой, классификация групп голономии отсутствует. Имеют место только частные результаты [22], [26], [27], [30], [58], [65], [69], [73], [85].…”

Holonomy Classification of Lorentz-Kähler Manifolds

“…The case of Lorentzian manifolds took the attention of geometers and theoretical physicists during the last two decades, see the reviews [3,25,26] and the references therein. In the other signatures, only partial results are known [12,13,7,23,24,28,29,35]. The main difference between holonomy groups of Riemannian and proper pseudo-Riemannian manifolds is that for Riemannian manifolds all considerations may be reduced to the case of irreducible holonomy groups, while in the case of proper pseudo-Riemannian manifolds one should consider also holonomy groups preserving degenerate subspaces.…”

Holonomy algebras of Einstein pseudo-Riemannian manifolds

“…The general case cannot be reduced to the irreducible one unless the metric is of the Riemannian signature. For pseudo-Riemannian manifolds of signature different from Riemannian and Lorentzian ones, only some partial case are considered [6,7,11,12,16,19,21,24]. There is also a classification of connected irreducible holonomy groups of torsion-free affine connections [26]; the groups corresponding to the Ricci-flat case are found in [4].…”