HYPERCOMPLEX STRUCTURES ON 4-DIMENSIONAL LIE-GROUPS

The purpose of this paper is to classify invariant hypercomplex struct ures on a 4-dimensional real Lie group G. It is shown that the 4-dimen sional simply connected Lie groups which admit invariant hypercomplex structures are the additive group H of the quaternions, the multiplica tive group H of nonzero quaternions, the solvable Lie groups acting s imply transitively on the real and complex hyperbolic spaces, RH4 and CH2, respectively, and the semidirect product C times sign with bar co nnected 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-m anifolds are determined.