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 ${\displaystyle G\times G\to G,(x,y)\mapsto xy^{-1}}$ is holomorphic. Basic examples are ${\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}$, 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 ${\displaystyle \mathbb {C} ^{*}}$). 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.