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 is holomorphic, 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 (in particular, complex Lie algebra) is a complex Lie group in an obvious way.

• A connected compact complex Lie group A of dimension g, a complex torus, where L is a surjective morphism of complex Lie groups, showing A is of the form described.

• , this is also an example of a representation of a complex Lie group that is not algebraic.

• Let X of holomorphic vector fields on X:.

• Let K be a connected compact Lie group. Then there exists a unique connected complex Lie group G, and (ii) K is a maximal compact subgroup of G. It is called the complexification of K 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: be the ring of holomorphic functions f on G spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G is the linear algebraic group that, when viewed as a complex manifold, is the original G of G.

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