Fine Art


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.

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.


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

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World