Complex Lie group explained

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 x G \to G, (x, y) \mapsto x y ^{-1} is holomorphic. Basic examples are \operatorname{GL} _{n(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 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

See also: Table of Lie groups.

• 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

C ^{g/L} , a complex torus, where L is a discrete subgroup of rank 2g. Indeed, its Lie algebra ak{a} can be shown to be abelian and then \operatorname{exp}: ak{a} \to A is a surjective morphism of complex Lie groups, showing A is of the form described.C \to 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 C ^{*} = \operatorname{GL} _{1(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) ⊗ _{R} C , and (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, \operatorname{GL} _{n(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]

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 ⋅ f spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G acts by left translation: g ⋅ 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) .

Notes and References

1. Guillemin. Victor. Sternberg. Shlomo. Geometric quantization and multiplicities of group representations. Inventiones Mathematicae. 1982. 67. 3. 515–538. 10.1007/bf01398934. 1982InMat..67..515G . 121632102 .