The holonomy of the Obata connection on Joyce hypercomplex manifolds

Abstract We study the holonomy of the Obata connection on Joyce hypercomplex manifolds. For all such group manifolds except $ \textrm{SU}(2n+1) $, we show that the holonomy group is strictly contained in the quaternionic general linear group. The case of $ \textrm{SU}(2n+1) $ is more subtle: for every $ n> 1 $, we show that there exist infinitely many Joyce hypercomplex structures with Obata holonomy strictly contained in $ \textrm{GL}(n(n+1),\mathbb{H}) $. On the other hand, Soldatenkov showed that $ \textrm{SU}(3) $ has Obata holonomy equal to $ \textrm{GL}(2, \mathbb{H}) $ [21], and we present here a new example on $ \textrm{SU}(5)$ with holonomy equal to $ \textrm{GL}(6,\mathbb{H}) $. Finally, we investigate Joyce hypercomplex manifolds whose restricted holonomy lies in $ \textrm{SL}(n, \mathbb{H}) $, yielding new compact examples of twisted Calabi–Yau manifolds.