Complex Lie group

In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it 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 a linear algebraic group.

The Lie algebra of a complex Lie group is a complex Lie algebra.

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 $$\mathbb{C}^g/L$$, a complex torus, where L is a discrete subgroup of rank 2g. 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 surjective homomorphism 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, analogous to the real case, $$\operatorname{Aut}(X)$$ is a complex Lie group whose Lie algebra is the space $$\Gamma(X, TX)$$ of holomorphic vector fields on X:.
 * Let K be a connected compact Lie group. Then there exists a unique connected complex Lie group G such that (i) $$\operatorname{Lie} (G) = \operatorname{Lie} (K) \otimes_{\mathbb{R}} \mathbb{C}$$, and (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.

Linear algebraic group associated to a complex semisimple Lie group
Let G be a complex semisimple Lie group. Then G admits a natural structure of a linear algebraic group as follows: let $$A$$ be the ring of holomorphic functions f on G such that $$G \cdot f$$ spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G acts by left translation: $$g \cdot f(h) = f(g^{-1}h)$$). Then $$\operatorname{Spec}(A)$$ is the linear algebraic group that, when viewed as a complex manifold, is the original G. More concretely, choose a faithful representation $$\rho : G \to GL(V)$$ of G. Then $$\rho(G)$$ is Zariski-closed in $$GL(V)$$.