Holonomy of the Obata connection on SU(3)

TL;DR

This paper investigates the holonomy group of the Obata connection on the Lie group SU(3), establishing that it equals GL(2, H), thus advancing understanding of hypercomplex structures on Lie groups.

Contribution

It proves that the holonomy of the Obata connection on SU(3) is exactly GL(2, H), providing a specific example of hypercomplex structure holonomy.

Findings

Holonomy of Obata connection on SU(3) is GL(2, H)

Supports Joyce's construction of hypercomplex structures on Lie groups

Enhances understanding of hypercomplex structures' holonomy groups

Abstract

A hypercomplex structure on a smooth manifold is a triple of integrable almost complex structures satisfying quaternionic relations. The Obata connection is the unique torsion-free connection that preserves each of the complex structures. The holonomy group of the Obata connection is contained in $GL(n, \mathbb{H})$. There is a well-known construction of hypercomplex structures on Lie groups due to Joyce. In this paper we show that the holonomy of the Obata connection on SU(3) coincides with $GL(2, \mathbb{H})$.