Complex structures on affine motion groups

Abstract

We study existence of complex structures on semidirect products g ⊕ρ v, where g is a real Lie algebra and ρ is a representation of g on v. Our first examples, the Euclidean algebra e(3) and the Poincaré algebra e(2, 1), carry complex structures obtained by deformation of a regular complex structure on sl (2, C). We also exhibit a complex structure on the Galilean algebra G(3, 1). We construct next a complex structure on g ⊕ρ v starting with one on g under certain compatibility assumptions on ρ. As an application of our results we obtain that there exists k ∈ {0, 1} such that (S1)k × E(n) admits a left invariant complex structure, where S1 is the circle and E(n) denotes the Euclidean group. We also prove that the Poincaré group P4k+3 has a natural left invariant complex structure. In case dim g = dim v, there is an adapted complex structure on g ⊕ρ v precisely when ρ determines a flat, torsion-free connection on g. If ρ is self-dual, g ⊕ρ v carries a natural symplectic structure as well. If, moreover, ρ comes from a metric connection then g ⊕ρ v possesses a pseudo-Kähler structure. We prove that the tangent bundle TG of a Lie group G carrying a flat torsion-free connection ∇ and a parallel complex structure possesses a hypercomplex structure. More generally, by an iterative procedure, we can obtain Lie groups carrying a family of left invariant complex structures which generate any prescribed real Clifford algebra.