Hypercomplex structures on four-dimensional Lie groups

The purpose of this paper is to classify invariant hypercomplex structures on a $4$-dimensional real Lie group $G$. It is shown that the $4$-dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group $\mathbb H$ of the quaternions, the multiplicative group ${\mathbb H}^*$ of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, ${\mathbb R}H^4$ and ${\mathbb C}H^2$, respectively, and the semidirect product ${\mathbb C}\rtimes {\mathbb C}$. We show that the spaces ${\mathbb C}H^2$ and ${\mathbb C}\rtimes {\mathbb C}\,$ possess an ${\mathbb R}P^2$ of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian $4$-manifolds are determined.