Complex structures on affine motion groups

Abstract

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 Poincar\'e 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 $k\in \{0,1\}$ such that $(S^1)^k \times E(n)$ admits a left invariant complex structure, where $S^1$ is the circle and E(n) denotes the Euclidean group. We also prove that the Poincar\'e group $P^{4k+3}$ has a natural left invariant complex structure. In case $\dim \g= \dim \v$, then 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-K\"ahler 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.