Complex Lie Groups

  • Joachim HilgertEmail author
  • Karl-Hermann Neeb
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter, we discuss complex Lie groups. Since we did not go into the theory of complex manifolds, we do this in a quite pedestrian fashion, but this will be enough for our purposes which are of a group theoretic nature. In particular, we define a complex Lie group as a real Lie group G whose Lie algebra \(\mathop {\bf L{}}\nolimits (G)\) is a complex Lie algebra and for which the adjoint representation maps into the group \(\mathop {\rm Aut}\nolimits _{{\mathbb{C}}}(\mathop {\bf L{}}\nolimits (G))\) of complex linear automorphisms of \(\mathop {\bf L{}}\nolimits (G)\) (the latter condition is automatically satisfied if G is connected). One can show that this is equivalent to G carrying the structure of a complex manifold such that the group operations are holomorphic, but we shall never need this additional structure.


Maximal Torus Complex Vector Space Holomorphic Representation Group Theoretic Nature 
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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of PaderbornPaderbornGermany
  2. 2.Department of MathematicsFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

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