Complex and Cr-structures on Compact Lie Groups Associated to Abelian Actions
نویسندگان
چکیده
It was shown by Samelson [9] and Wang [10] 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 [2] 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-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.
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