Revisiting the classification of homogeneous 3-Sasakian and quaternionic Kähler manifolds

Abstract

AbstractWe provide a new, self-contained proof of the classification of homogeneous 3-Sasakian manifolds, which was originally obtained by Boyer et al. (J Reine Angew Math 455:183–220, [10]). 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 Kähler manifolds due to Alekseevskii (Funct Anal Appl 2(2):106–114, [2]) from our results.