Complex Lie group

Examples

• A finite-dimensional vector space over the complex numbers is a complex Lie group in an obvious way.
• A connected compact complex Lie group A of dimension g is of the form where L is a discrete subgroup. Indeed, its Lie algebra can be shown to be abelian and then is a surjective morphism of complex Lie groups, showing A is of the form described.
• is an example of a morphism of complex Lie groups that does not come from a morphism of algebraic groups. Since, 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, is a complex Lie group whose Lie algebra is.
• Let K be a connected compact Lie group. Then there exists a unique connected complex Lie group G such that , and K is a maximal compact subgroup of G. It is called the complexification of K. For example, 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