Hypercomplex structures on special linear groups

The purpose of this article is twofold. First, we prove that the 8-dimensional Lie group SL (3, R) does not admit a left-invariant hypercomplex structure. To accomplish this we revise the classification of left-invariant complex structures on SL (3, R) due to Sasaki. Second, we exhibit a left-invariant hypercomplex structure on SL (2n+1, C), which arises from a complex product structure on SL (2n+1, R), for all n N. We then show that there are no HKT metrics compatible with this hypercomplex structure. Additionally, we determine the associated Obata connection and we compute explicitly its holonomy group, providing thus a new example of an Obata holonomy group properly contained in GL (m, H) and not contained in SL (m, H), where 4m=R SL (2n+1, C).