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 is holomorphic. Basic examples are, the general linear groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus. 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 is a complex Lie group in an obvious way.
- A connected compact complex Lie group A of dimension g is of the form, a complex torus, where L is a discrete subgroup of rank 2g. 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 surjective homomorphism 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, analogous to the real case, is a complex Lie group whose Lie algebra is the space of holomorphic vector fields on X:.
- 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