Hyper-Kaehler Metrics Conformal to Left Invariant Metrics on Four-Dimensional Lie Groups

Let g be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold (M,H). We show that when the isometry group I(M,g) contains a subgroup G acting simply transitively on M by hypercomplex isometries, then the metric g is conformal to a hyper-Kaehler metric. We describe explicitely the corresponding hyper-Kaehler metrics, which are of cohomegeneity one with respect to a 3-dimensional normal subgroup of G. It follows that, in four dimensions, these are the only hyper-Kaehler metrics containing a homogeneous metric in its conformal class