Revisiting the Classification of Homogeneous 3-Sasakian and Quaternionic K\"ahler Manifolds

TL;DR

This paper offers a new proof for classifying homogeneous 3-Sasakian manifolds, linking them to simple complex Lie algebras, and explains their connection to quaternionic K"ahler manifolds, providing insights into their structure and classification.

Contribution

It provides a self-contained proof of the classification, constructs an explicit correspondence with Lie algebras, and clarifies the structure of non-simply connected cases.

Findings

Explicit correspondence between 3-Sasakian manifolds and Lie algebras

Real projective spaces are the only non-simply connected cases

Derivation of the quaternionic K"ahler classification from the results

Abstract

We provide a new, self-contained proof of the classification of homogeneous 3-Sasakian manifolds, which was originally obtained by Boyer, Galicki and Mann. In doing so, we construct an explicit one-to-one correspondence between simply connected homogeneous 3-Sasakian manifolds and simple complex Lie algebras via the theory of root systems. We also discuss why the real projective spaces are the only non-simply connected homogeneous 3-Sasakian manifolds and derive the famous classification of homogeneous positive quaternionic K\"ahler manifolds due to Wolf and Alekseevskii from our results.