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

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.