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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.

Examples

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.

References

Lee, Dong Hoon (2002), The Structure of Complex Lie Groups (PDF), Boca Raton, FL: Chapman & Hall/CRC, ISBN 1-58488-261-1, MR 1887930
Serre, Jean-Pierre (1993), Gèbres