Holonomy of the Obata connection on SU(3)

Abstract

A hypercomplex structure on a smooth manifold is a triple of integrable almost complex structures satisfying quaternionic relations. The Obata connection is the unique torsion-free connection that preserves each of the complex structures. The holonomy group of the Obata connection is contained in $GL(n, \mathbb{H})$. There is a well-known construction of hypercomplex structures on Lie groups due to Joyce. In this paper we show that the holonomy of the Obata connection on SU(3) coincides with $GL(2, \mathbb{H})$.