# Complex Lie group

In geometry, a **complex Lie group** is a complex-analytic manifold that is also a group in such a way is holomorphic. Basic examples are , 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 ). 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 where*L*is a discrete subgroup. Indeed, its Lie algebra can be shown to be abelian and then is a surjective morphism of complex Lie groups, showing*A*is of the form described. - is an example of a morphism of complex Lie groups that does not come from a morphism of algebraic groups. Since , 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, is a complex Lie group whose Lie algebra is . - Let
*K*be a connected compact Lie group. Then there exists a unique connected complex Lie group*G*such that (i) (ii)*K*is a maximal compact subgroup of*G*. It is called the complexification of*K*. For example, 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*.^{[1]}

## References

- ↑ Guillemin, Victor; Sternberg, Shlomo (1982). "Geometric quantization and multiplicities of group representations".
*Inventiones Mathematicae*.**67**(3): 515–538.

- 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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