Holonomy of the Obata connection on \(SU(3)\) (Q2909342)

A smooth manifold with a triple of almost complex structures satisfying quaternonic relations is called hypercomplex. (A special case are the hyper-Kähler manifolds.) Any hypercomplex manifold is equipped with a distinguished torsion-free connection preserving all the complex structures, the so-called Obata connection. Its holonomy group is a significant invariant for such manifolds. A priori this holonomy group sits in \(Gl(n,\mathbb{H})\), which is one of the possible irreducible holonomy groups of non-metric type for torsion-free connections.NEWLINENEWLINESo far it was unclear whether the full group \(Gl(n,\mathbb{H})\) can occur as holonomy of a compact hypercomplex manifold. In the current paper the author shows the Lie group \(SU(3)\) admits such a hypercomplex structure. This construction follows a general principle of Joyce for homogeneous hypercomplex structures on compact Lie groups. In particular, \(SU(3)\) admits an Euler vector, which implies that only tensors that are preserved by the Obata connection must be of type \((k,k)\). Further, computing curvature terms and using the Ambrose-Singer theorem, the author can finally conclude that the holonomy is irreducible on \(SU(3)\). Comparing with the complete list of non-metric holonomy groups, the only possibility that remains is \(Gl(2,\mathbb{H})\).