Holonomy algebras of pseudo-hyper-K\"ahlerian manifolds of index 4

TL;DR

This paper classifies the possible holonomy algebras of pseudo-hyper-K"ahlerian manifolds with signature (4,4n+4) and provides a new proof for the classification of their symmetric spaces of index 4.

Contribution

It offers a complete classification of holonomy algebras for these manifolds and introduces a novel proof for symmetric space classification.

Findings

Holonomy algebras are subalgebras of sp(1,n+1).

Classification of possible holonomy algebras is achieved.

New proof of symmetric space classification of index 4 is provided.

Abstract

The holonomy algebra of a pseudo-hyper-K\"ahlerian 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-K\"ahlerian symmetric spaces of index 4 is obtained.