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 H of the quaternions, the multiplicative group H* of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, RH4 and CH2, respectively, and the semidirect product C times sign with bar connected to right of it C. We show that the spaces CH2 and C times sign with bar connected to right of it C possess an RP2 of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian 4-manifolds are determined.