Hypercomplex structures on special linear groups

TL;DR

This paper investigates hypercomplex structures on special linear groups, proving non-existence on certain groups, constructing examples on others, and analyzing associated geometric properties and holonomy groups.

Contribution

It proves the non-existence of left-invariant hypercomplex structures on SL(3,R) and constructs such structures on SL(2n+1,C), analyzing their geometric and holonomy properties.

Findings

SL(3,R) does not admit a left-invariant hypercomplex structure

A hypercomplex structure exists on SL(2n+1,C) derived from a complex product structure

The associated Obata connection has a holonomy group properly contained in GL(m,H) but not in SL(m,H)

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 $4m=\dim_\mathbb{R} \operatorname{SL}(2n+1,\mathbb{C})$.