nLab complex Lie group

Idea

A complex Lie group is a Lie group that is a group object not just internal to smooth manifolds but in fact to complex manifolds. Hence it is a complex manifold $G$ equipped with a group structure such that both the multiplication map $G \times G \to G$ as well as the inverse map $G \to G$ are holomorphic functions.

Properties

Relation to almost complex structure

One can also consider groups in almost complex manifolds. But every almost complex Lie group is automatically also a complex Lie group.

Complexification of compact Lie groups

For $K$ a compact Lie group there is a unique connected complex Lie group $G$ such that

1. the Lie algebra of $G$ is the complexification of the Lie algebra of $K$:

   $$Lie(G) \simeq Lie(K) \otimes_{\mathbb{R}} \mathbb{C} \,,$$

2. $K$ is a maximal compact subgroup of $G$.

This $G$ is called the complexification of $K$.

Oka property

Proposition

(coset spaces of complex Lie groups are Oka manifolds)
Every complex Lie group and every coset space (homogeneous space) of complex Lie groups is an Oka manifold.

(review in Forstnerič & Lárusson 11, p. 9, Forstnerič 2013, Thm. 2.6)

Related concepts

• complex Lie algebra

• relation between compact Lie groups and reductive algebraic groups

References

• Dietmar Salamon, Notes on complex Lie groups, 2019 (pdf, pdf)

See also

• Wikipedia, Complex Lie group