Homogeneous Hypercomplex Structures and the Joyce's Construction

TL;DR

This paper demonstrates that all invariant hypercomplex structures on certain homogeneous spaces can be constructed using Joyce's method, assuming the existence of a compatible hyper-Hermitian metric.

Contribution

It establishes a comprehensive link between invariant hypercomplex structures and Joyce's construction under specific metric conditions.

Findings

All invariant hypercomplex structures on G/L are obtainable via Joyce's construction.

Existence of a hyper-Hermitian naturally reductive metric is key to the construction.

The result applies to homogeneous spaces with compact Lie groups.

Abstract

We prove that any invariant hypercomplex structure on a homogeneous space $M = G/L$ where $G$ is a compact Lie group is obtained via the Joyce's construction, provided that there exists a hyper-Hermitian naturally reductive invariant metric on $M$.