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 (not to be confused with the complex Lie group \mathbb C^*). 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

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: let A be the ring of holomorphic functions f on G such that G \cdot f spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G acts by left translation: ). Then is the linear algebraic group that, when viewed as a complex manifold, is the original G. More concretely, choose a faithful representation of G. Then \rho(G) is Zariski-closed in GL(V).