Hypercomplex structures on special linear groups
Adrián Andrada, Agustín Garrone, Alejandro Tolcachier
TL;DR
The paper investigates left-invariant hypercomplex structures on special linear groups by tying Lie-algebraic complex structures to hypercomplex geometry. It proves non-existence for $SL(3,\mathbb{R})$ via a corrected Sasaki classification, and constructs left-invariant hypercomplex structures on $SL(2n+1,\mathbb{C})$ via a complex product framework derived from a complexification of $SL(2n+1,\mathbb{R})$, while showing no compatible left-invariant HKT metric. It further computes the Obata connection holonomy for the constructed hypercomplex structure, obtaining an explicit holonomy algebra $\mathfrak{hol}^{\mathrm{Ob}} \cong \mathfrak{gl}(n,\mathbb{C}) \ltimes_{\mu^{\mathbb{C}}_{2n+1}} V^{\mathbb{C}}_{2n+1}$, which is not contained in $\mathrm{GL}(m,\mathbb{H})$ or $\mathrm{SL}(m,\mathbb{H})$, providing a new, concrete example outside the standard quaternionic holonomy classes. The work highlights how complex-product methods on real forms yield hypercomplex structures on complexifications and informs Obata holonomy theory for non-HKT hypercomplex Lie groups.
Abstract
The purpose of this article is twofold. First, we prove that the $8$-dimensional Lie group $\operatorname{SL}(3,\mathbb{R})$ does not admit a left-invariant hypercomplex structure. To accomplish this we revise the classification of left-invariant complex structures on $\operatorname{SL}(3,\mathbb{R})$ due to Sasaki. Second, we exhibit a left-invariant hypercomplex structure on $\operatorname{SL}(2n+1,\mathbb{C})$, which arises from a complex product structure on $\operatorname{SL}(2n+1,\mathbb{R})$, 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 $\operatorname{GL}(m,\mathbb{H})$ and not contained in $\operatorname{SL}(m,\mathbb{H})$, where $4m=\dim_\mathbb{R} \operatorname{SL}(2n+1,\mathbb{C})$.