Complex Lie group

<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a complex Lie group is a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> over the complex numbers; i.e., it is a <a href="/facts/Complex_manifold/DlvPnpzm"> complex-analytic manifold</a> that is also a <a href="/facts/Group_(mathematics)/IQTYqINt"> group</a> in such a way 
  
    
      
        G
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    {\displaystyle G\times G\to G,(x,y)\mapsto xy^{-1}}
  
 is <a href="/facts/Holomorphic/kVzthtEf">holomorphic</a>. Basic examples are 
  
    
      
        
          GL
          
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    {\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}
  
, the <a href="/facts/General_linear_group/B54gNILS">general linear groups</a> over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>. A connected compact complex Lie group is precisely a <a href="/facts/Complex_torus/nh9BNgYU">complex torus</a> (not to be confused with the complex Lie group 
  
    
      
        
          
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    {\displaystyle \mathbb {C} ^{*}}
  
). Any finite group may be given the structure of a complex Lie group. A complex <a href="/facts/Semisimple_Lie_group/m8jA3Bcf">semisimple Lie group</a> is a <a href="/facts/Linear_algebraic_group/eMLtu5Nh">linear algebraic group</a>.
</p><p>The Lie algebra of a complex Lie group is a <a href="/facts/Complex_Lie_algebra/qCcze5of">complex Lie algebra</a>.
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Complex Lie group open-in-new

Complex Lie group