Complex Lie algebra

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a complex Lie algebra is a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>.
</p><p>Given a complex Lie algebra 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
  
, its conjugate 
  
    
      
        
          
            
              g
            
            ¯
          
        
      
    
    {\displaystyle {\overline {\mathfrak {g}}}}
  
 is a complex Lie algebra with the same underlying <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> but with 
  
    
      
        i
        =
        
          
            −
            1
          
        
      
    
    {\displaystyle i={\sqrt {-1}}}
  
 acting as 
  
    
      
        −
        i
      
    
    {\displaystyle -i}
  
 instead. As a real Lie algebra, a complex Lie algebra 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
  
 is trivially <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to its conjugate. A complex Lie algebra is isomorphic to its conjugate <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it admits a real form (and is said to be defined over the real numbers).
</p>

Complex Lie algebra open-in-new

Complex Lie algebra