Complex Lie group

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In geometry, a complex Lie group is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^{-1} is holomorphic. Basic examples are \operatorname{GL}_n(\mathbb{C}), the general linear groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus (not to be confused with the complex Lie group \mathbb C^*). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is an algebraic group.


  • A finite-dimensional vector space over the complex numbers (in particular, complex Lie algebra) is a complex Lie group in an obvious way.
  • A connected compact complex Lie group A of dimension g is of the form \mathbb{C}^g/L where L is a discrete subgroup. Indeed, its Lie algebra \mathfrak{a} can be shown to be abelian and then \operatorname{exp}: \mathfrak{a} \to A is a surjective morphism of complex Lie groups, showing A is of the form described.
  • \mathbb{C} \to \mathbb{C}^*, z \mapsto e^z is an example of a morphism of complex Lie groups that does not come from a morphism of algebraic groups. Since \mathbb{C}^* = \operatorname{GL}_1(\mathbb{C}), this is also an example of a representation of a complex Lie group that is not algebraic.
  • Let X be a compact complex manifold. Then, as in the real case, \operatorname{Aut}(X) is a complex Lie group whose Lie algebra is \Gamma(X, TX).
  • Let K be a connected compact Lie group. Then there exists a unique connected complex Lie group G such that (i) \operatorname{Lie} (G) = \operatorname{Lie} (K) \otimes_{\mathbb{R}} \mathbb{C} (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, \operatorname{GL}_n(\mathbb{C}) is the complexification of the unitary group. If K is acting on a compact kähler manifold X, then the action of K extends to that of G.[citation needed]