  It was shown by Samelson and Wang that each compact Lie group K of even
dimension admits left-invariant complex structures. When K has odd dimension it
admits a left-invariant CR-structure of maximal dimension. This has been proved
recently by Charbonnel and Khalgui who have also given a complete algebraic
description of these structures. In this article we present an alternative and
more geometric construction of this type of invariant structures on a compact
Lie group K when it is semisimple. We prove that each left-invariant complex
structure, or each CR-structure of maximal dimension with a transverse
CR-action by R, is induced by a holomorphic C^l action on a quasi-projective
manifold X naturally associated to K. We then show that X admits more general
Abelian actions, also inducing complex or CR structures on K which are
generically non-invariant.
