Complex Lie group

In geometry, a complex Lie group is a complex-analytic manifold that is also a group in such a way G × G → G , ( x , y ) ↦ x y − 1 {displaystyle G imes G o G,(x,y)mapsto xy^{-1}} is holomorphic. Basic examples are GL n ⁡ ( C ) {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 C ∗ {displaystyle mathbb {C} ^{*}} ). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is an algebraic group. In geometry, a complex Lie group is a complex-analytic manifold that is also a group in such a way G × G → G , ( x , y ) ↦ x y − 1 {displaystyle G imes G o G,(x,y)mapsto xy^{-1}} is holomorphic. Basic examples are GL n ⁡ ( C ) {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 C ∗ {displaystyle mathbb {C} ^{*}} ). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is an algebraic group.