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.