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The Quarterly Journal of Mathematics 2004 55(4):375-389; doi:10.1093/qmath/hah013
© 2004 by Oxford University Press
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Complex structures on affine motion groups

María L. Barberis * and Isabel G. Dotti §

CIEM, FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, (5000) Córdoba, Argentina  *

We study existence of complex structures on semidirect products g {oplus}{rho} v, where g is a real Lie algebra and {rho} 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 {oplus}{rho} v starting with one on g under certain compatibility assumptions on {rho}.

As an application of our results we obtain that there exists k {0, 1} such that (S1)k x 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 {oplus}{rho} v precisely when {rho} determines a flat, torsion-free connection on g. If {rho} is self-dual, g {oplus}{rho} v carries a natural symplectic structure as well. If, moreover, {rho} comes from a metric connection then g {oplus}{rho} v possesses a pseudo-Kähler structure.

We prove that the tangent bundle TG of a Lie group G carrying a flat torsion-free connection {nabla} 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.

Received 21 July 2003.

* E-mail: barberis{at}mate.uncor.edu

§ E-mail: idotti{at}mate.uncor.edu


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