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 $G\times G\to G,(x,y)\mapsto xy^{-1}$ 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 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 $\mathbb {C} ^{g}/L$ 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) $\operatorname {Lie} (G)=\operatorname {Lie} (K)\otimes _{\mathbb {R} }\mathbb {C}$ , 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.^[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:^[2] 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 $GL(V)$ .^{[clarification needed]}

References

^ Guillemin, Victor; Sternberg, Shlomo (1982). "Geometric quantization and multiplicities of group representations". Inventiones Mathematicae. 67 (3): 515–538. doi:10.1007/bf01398934.
^ Serre & Ch. VIII. Theorem 10.

Lee, Dong Hoon (2002), The Structure of Complex Lie Groups (PDF), Boca Raton, Florida: Chapman & Hall/CRC, ISBN 1-58488-261-1, MR 1887930^{[permanent dead link]}
Serre, Jean-Pierre (1993), Gèbres^{[permanent dead link]}

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

[2] Serre & Ch. VIII. Theorem 10.

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