On the non-existence of left-invariant hypercomplex structures on $SU(2)^{4n}$

TL;DR

This paper proves that the compact Lie groups formed by products of SU(2) do not admit left-invariant hypercomplex structures, clarifying a previously confusing aspect in the literature.

Contribution

It provides a simple algebraic proof that no product of SU(2) groups admits such structures, resolving a longstanding open question.

Findings

SU(2)^m has no left-invariant hypercomplex structures for all m≥1

Clarifies the non-existence of hypercomplex structures on certain compact Lie groups

Resolves confusion in recent mathematical literature

Abstract

Using elementary algebraic arguments, it is shown that $SU(2)^{m}:=SU(2)\times \cdots \times SU(2)$ ($m$ times) admits no left-invariant hypercomplex structures for all $m\ge 1$. This result answers (in a clear and easily accessible way) the question of whether every compact Lie group of dimension $4n$ admits a left-invariant hypercomplex structure. The aforementioned question has apparently been the source of some confusion in the recent literature.