Invariant complex structures on tangent and cotangent Lie groups of dimension six

Abstract

This paper deals with left invariant complex structures on simply connected Lie groups, the Lie algebra of which is of the type $\mathrm{T}_{\pi} \mathfrak{h}=\mathfrak{h} \ltimes_{\pi} V$, where $\pi$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\mathrm{T}_{\pi} \mathfrak{h}$ for $\mathfrak{h}$ a three dimensional real Lie algebra. First it was proposed the study of complex structures $J$ satisfying the constraint $J\mathfrak{h} = V$. Whenever $\pi$ is the adjoint representation this kind of complex structures are associated to non-singular derivations of $\mathfrak{h}$. This fact allows different kinds of applications.