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Anyonic Lie algebra

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an <a href="/facts/Anyon/hfgpCFjf">anyonic</a> <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> is a <i>U</i>(1) <a href="/facts/Graded_vector_space/Ff1maywV">graded vector space</a> 
  
    
      
        L
      
    
    {\displaystyle L}
  
 over 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
 equipped with a <a href="/facts/Bilinear_operator/LH1lHVeP">bilinear operator</a> 
  
    
      
        [
        ⋅
        ,
        ⋅
        ]
        :
        L
        ×
        L
        →
        L
      
    
    {\displaystyle [\cdot ,\cdot ]\colon L\times L\rightarrow L}
  
 and <a href="/facts/Linear_map/Q1PBKNVV">linear maps</a> 
  
    
      
        ε
        :
        L
        →
        
          C
        
      
    
    {\displaystyle \varepsilon \colon L\to \mathbb {C} }
  
 (some authors use 
  
    
      
        
          |
        
        ⋅
        
          |
        
        :
        L
        →
        
          C
        
      
    
    {\displaystyle |\cdot |\colon L\to \mathbb {C} }
  
) and 
  
    
      
        Δ
        :
        L
        →
        L
        ⊗
        L
      
    
    {\displaystyle \Delta \colon L\to L\otimes L}
  
 such that 
  
    
      
        Δ
        X
        =
        
          X
          
            i
          
        
        ⊗
        
          X
          
            i
          
        
      
    
    {\displaystyle \Delta X=X_{i}\otimes X^{i}}
  
, satisfying following axioms:
</p>
<ul><li>
  
    
      
        ε
        (
        [
        X
        ,
        Y
        ]
        )
        =
        ε
        (
        X
        )
        ε
        (
        Y
        )
      
    
    {\displaystyle \varepsilon ([X,Y])=\varepsilon (X)\varepsilon (Y)}
  
</li>
<li>
  
    
      
        [
        X
        ,
        Y
        
          ]
          
            i
          
        
        ⊗
        [
        X
        ,
        Y
        
          ]
          
            i
          
        
        =
        [
        
          X
          
            i
          
        
        ,
        
          Y
          
            j
          
        
        ]
        ⊗
        [
        
          X
          
            i
          
        
        ,
        
          Y
          
            j
          
        
        ]
        
          e
          
            
              
                
                  2
                  π
                  i
                
                n
              
            
            ε
            (
            
              X
              
                i
              
            
            )
            ε
            (
            
              Y
              
                j
              
            
            )
          
        
      
    
    {\displaystyle [X,Y]_{i}\otimes [X,Y]^{i}=[X_{i},Y_{j}]\otimes [X^{i},Y^{j}]e^{{\frac {2\pi i}{n}}\varepsilon (X^{i})\varepsilon (Y_{j})}}
  
</li>
<li>
  
    
      
        
          X
          
            i
          
        
        ⊗
        [
        
          X
          
            i
          
        
        ,
        Y
        ]
        =
        
          X
          
            i
          
        
        ⊗
        [
        
          X
          
            i
          
        
        ,
        Y
        ]
        
          e
          
            
              
                
                  2
                  π
                  i
                
                n
              
            
            ε
            (
            
              X
              
                i
              
            
            )
            (
            2
            ε
            (
            Y
            )
            +
            ε
            (
            
              X
              
                i
              
            
            )
            )
          
        
      
    
    {\displaystyle X_{i}\otimes [X^{i},Y]=X^{i}\otimes [X_{i},Y]e^{{\frac {2\pi i}{n}}\varepsilon (X_{i})(2\varepsilon (Y)+\varepsilon (X^{i}))}}
  
</li>
<li>
  
    
      
        [
        X
        ,
        [
        Y
        ,
        Z
        ]
        ]
        =
        [
        [
        
          X
          
            i
          
        
        ,
        Y
        ]
        ,
        [
        
          X
          
            i
          
        
        ,
        Z
        ]
        ]
        
          e
          
            
              
                
                  2
                  π
                  i
                
                n
              
            
            ε
            (
            Y
            )
            ε
            (
            
              X
              
                i
              
            
            )
          
        
      
    
    {\displaystyle [X,[Y,Z]]=[[X_{i},Y],[X^{i},Z]]e^{{\frac {2\pi i}{n}}\varepsilon (Y)\varepsilon (X^{i})}}
  
</li></ul>
<p>for pure graded elements <i>X</i>, <i>Y</i>, and <i>Z</i>.
</p>