# 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

• 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

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