Complex structures on affine motion groups

M. L. BARBERIS, I. DOTTI

QUARTERLY JOURNAL OF MATHEMATICS

Oxford University Press

Lugar: Oxford; Año: 2004 vol. 55 p. 375 - 389

0033-5606

We study existence of complex structures on semidirect products  g x r v where g is a real Lie algebra and r 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 x r v starting with one on g under certain compatibility assumptions on  r. As an application of our results we obtain that there exists k in {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,  then there is an adapted complex structure on g x r v  precisely when r determines a    flat,  torsion-free connection on g. If r is self-dual, g x r v carries a natural symplectic structure as well. If, moreover, r comes from a metric connection then g x r v  possesses a pseudo-Kähler structure. We prove that the tangent bundle TG of a Lie group G carrying a flat torsion free connection D 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.