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

, 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

Any finite group may be given the structure of a complex Lie group.

A complex semisimple Lie group is a linear algebraic group.

Let G be a complex semisimple Lie group.

Then G admits a natural structure of a linear algebraic group as follows:[2] let

be the ring of holomorphic functions f on G such that

spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G acts by left translation:

is the linear algebraic group that, when viewed as a complex manifold, is the original G. More concretely, choose a faithful representation

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