Special linear Lie algebra

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the special linear Lie algebra of order 
  
    
      
        n
      
    
    {\displaystyle n}
  
 over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
  
    
      
        F
      
    
    {\displaystyle F}
  
, denoted 
  
    
      
        
          
            
              s
              l
            
          
          
            n
          
        
        F
      
    
    {\displaystyle {\mathfrak {sl}}_{n}F}
  
 or 
  
    
      
        
          
            s
            l
          
        
        (
        n
        ,
        F
        )
      
    
    {\displaystyle {\mathfrak {sl}}(n,F)}
  
, is the <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> of all the 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> (with entries in 
  
    
      
        F
      
    
    {\displaystyle F}
  
) with <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a> zero and with the <a href="/facts/Lie_algebra/n2gAUkNq">Lie bracket</a> 
  
    
      
        [
        X
        ,
        Y
        ]
        :=
        X
        Y
        −
        Y
        X
      
    
    {\displaystyle [X,Y]:=XY-YX}
  
 given by the <a href="/facts/Commutator/eYxzFmFY">commutator</a>. This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras.  The <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> that it generates is the <a href="/facts/Special_linear_group/rsahah9w">special linear group</a>.
</p>

Special linear Lie algebra open-in-new

Special linear Lie algebra