# Complex Lie group

In geometry, a complex Lie group is a complex-analytic manifold that is also a group in such a way ${\displaystyle G\times G\to G,(x,y)\mapsto xy^{-1}}$ is holomorphic. Basic examples are ${\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}$, 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 ${\displaystyle \mathbb {C} ^{*}}$). 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 ${\displaystyle \mathbb {C} ^{g}/L}$ where L is a discrete subgroup. Indeed, its Lie algebra ${\displaystyle {\mathfrak {a}}}$ can be shown to be abelian and then ${\displaystyle \operatorname {exp} :{\mathfrak {a}}\to A}$ is a surjective morphism of complex Lie groups, showing A is of the form described.
• ${\displaystyle \mathbb {C} \to \mathbb {C} ^{*},z\mapsto e^{z}}$ is an example of a morphism of complex Lie groups that does not come from a morphism of algebraic groups. Since ${\displaystyle \mathbb {C} ^{*}=\operatorname {GL} _{1}(\mathbb {C} )}$, 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, ${\displaystyle \operatorname {Aut} (X)}$ is a complex Lie group whose Lie algebra is ${\displaystyle \Gamma (X,TX)}$.
• Let K be a connected compact Lie group. Then there exists a unique connected complex Lie group G such that (i) ${\displaystyle \operatorname {Lie} (G)=\operatorname {Lie} (K)\otimes _{\mathbb {R} }\mathbb {C} }$ (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, ${\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}$ 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.[1]

## References

1. ^ Guillemin, Victor; Sternberg, Shlomo (1982). "Geometric quantization and multiplicities of group representations". Inventiones Mathematicae. 67 (3): 515–538. doi:10.1007/bf01398934.