Complex Lie group

{{Short description|Lie group whose manifold is complex and whose group operation is holomorphic}}

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:.{{clarify|date=March 2023}}
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.{{cite journal|last1=Guillemin|first1=Victor|last2=Sternberg|first2=Shlomo|title=Geometric quantization and multiplicities of group representations|journal=Inventiones Mathematicae|date=1982|volume=67|issue=3|pages=515–538|doi=10.1007/bf01398934|bibcode=1982InMat..67..515G |s2cid=121632102 }}

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:{{harvnb|Serre|1993|p=Ch. VIII. Theorem 10.}} 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 $GL(V)$ .{{clarify|why closed?|date=February 2020}}

References

{{citation

| last = Lee

| first = Dong Hoon

| isbn = 1-58488-261-1

| mr = 1887930

| publisher = Chapman & Hall/CRC

| location = Boca Raton, Florida

| title = The Structure of Complex Lie Groups

| year = 2002

}}

{{citation | last=Serre | first=Jean-Pierre | title=Gèbres | work=L'Enseignement Mathématique | url=https://www.e-periodica.ch/digbib/view?pid=ens-001:1993:39::15#232 | year=1993 | volume=39 | issue=1–2 | page=33 | doi=10.5169/seals-60413 }}

Category:Lie groups

Category:Manifolds