Complex Lie group - Alchetron, The Free Social Encyclopedia

In geometry, a complex Lie group is a complex-analytic manifold that is also a group in such a way G × G → G , ( x , y ) ↦ x y − 1 is holomorphic. Basic examples are 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 an algebraic group.

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 C g / L where L is a discrete subgroup. Indeed, its Lie algebra a can be shown to be abelian and then exp : a → A is a surjective morphism of complex Lie groups, showing A is of the form described.

C → C ∗ , z ↦ e z is an example of a morphism of complex Lie groups that does not come from a morphism of algebraic groups. Since C ∗ = 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, as in the real case, Aut ⁡ ( X ) is a complex Lie group whose Lie algebra is Γ ( X , T X ) .

Let K be a connected compact Lie group. Then there exists a unique connected complex Lie group G such that (i) Lie ⁡ ( G ) = Lie ⁡ ( K ) ⊗ R C (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, 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.

References

Complex Lie group Wikipedia

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