Complex Lie group

In geometry, a complex Lie group is a complex-analytic manifold that is also a group in such a way $G\times G\to G,(x,y)\mapsto xy^{-1}$ 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.
• 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.