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 $G \times G \to G, (x, y) \mapsto x y^$ is holomorphic. Basic examples are $\operatorname_n(\mathbb)$, 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 $\mathbb^g/L$ where ''L'' is a discrete subgroup. Indeed, its Lie algebra $\mathfrak$ can be shown to be abelian and then $\operatorname: \mathfrak \to A$ is a surjective morphism of complex Lie groups, showing ''A'' is of the form described.
* $\mathbb \to \mathbb^*, z \mapsto e^z$ is an example of a surjective homomorphism of complex Lie groups that does not come from a morphism of algebraic groups. Since $\mathbb^* = \operatorname_1(\mathbb)$, 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, $\operatorname(X)$ 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) $\operatorname (G) = \operatorname (K) \otimes_ \mathbb$, and (ii) ''K'' is a maximal compact subgroup of ''G''. It is called the complexification of ''K''. For example, $\operatorname_n(\mathbb)$ 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: $g \cdot f(h) = f(g^h)$). Then $\operatorname(A)$ 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 $GL(V)$.

References

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Lie groups
Manifolds

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