Hypercomplex structures on special linear groups

Abstract

The purpose of this article is twofold. First, we prove that the $8$-dimensional Lie group $\mathrm{SL}(3,\mathbb{R})$ does not admit a left-invariant hypercomplex structure. To accomplish this we revise the classification of left-invariant complex structures on $\mathrm{SL}(3,\mathbb{R})$ due to Sasaki. Second, we exhibit a left-invariant hypercomplex structure on $\mathrm{SL}(2n+1,\mathbb{C})$, which arises from a complex product structure on $\mathrm{SL}(2n+1,\mathbb{R})$, for all $n\in \mathbb{N}$. 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 $\mathrm{GL}(m,\mathbb{H})$ and not contained in $\mathrm{SL}(m,\mathbb{H})$, where $4m={\dim }_{\mathbb{R}}\mathrm{SL}(2n+1,\mathbb{C})$.

Final version. To appear in Collect. Math