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Complex Lie group
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
See also: Table of Lie groups
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
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Graduate Studies in Mathematics
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