Homogeneous Hypercomplex Structures I - the compact Lie groups

TL;DR

This paper introduces a novel root subset called 'the stem' to classify invariant hypercomplex structures on compact Lie groups, providing new algebraic tools and decompositions for understanding these geometric structures.

Contribution

It presents a complete classification of invariant hypercomplex structures on compact Lie groups using the concept of the stem and related algebraic decompositions.

Findings

Introduction of the stem subset of positive roots

Development of algebraic decompositions of Lie algebras

Complete classification of invariant hypercomplex structures

Abstract

We introduce a remarkable subset "the stem" of the set of positive roots of a reduced root system. The stem determines several interesting decompositions of the corresponding reductive Lie algebra. It gives also a nice simple three dimensional subalgebra and a "Cayley transform". In the present paper we apply the above devices to give a complete classification of invariant hypercomplex structures on compact Lie groups.