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 formula_1 is holomorphic. Basic examples are formula_2, 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 formula_3). 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.
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 formula_13 be the ring of holomorphic functions "f" on "G" such that formula_14 spans a finite-dimensional vector space inside the ring of holomorphic functions on "G" (here "G" acts by left translation: formula_15). Then formula_16 is the linear algebraic group that, when viewed as a complex manifold, is the original "G". More concretely, choose a faithful representation formula_17 of "G". Then formula_18 is Zariski-closed in formula_19.