Lie conformal algebra

<h2 id="definition-and-relation-to-lie-algebras">Definition and relation to Lie algebras</h2>
<p>A Lie algebra is defined to be a vector space with a <a href="/facts/Skew_symmetric/nw5CQeLN">skew symmetric</a> <a href="/facts/Bilinear_map/LH1lHVeP">bilinear</a> multiplication which satisfies the <a href="/facts/Jacobi_identity/C9HoyQ3z">Jacobi identity</a>. More generally, a Lie algebra is an object, 
  
    
      
        L
      
    
    {\displaystyle L}
  
 in the category of <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> (read: 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
-modules) with a <a href="/facts/Morphism/Kdpuyz3S">morphism</a>
</p>

[
        ⋅
        ,
        ⋅
        ]
        :
        L
        ⊗
        L
        →
        L
      
    
    {\displaystyle [\cdot ,\cdot ]:L\otimes L\rightarrow L}

<p>that is skew-symmetric and satisfies the Jacobi identity. A Lie conformal algebra, then, is an object 
  
    
      
        R
      
    
    {\displaystyle R}
  
 in the category of 
  
    
      
        
          C
        
        [
        ∂
        ]
      
    
    {\displaystyle \mathbb {C} [\partial ]}
  
-modules with morphism
</p>

[
        
          ⋅
          
            λ
          
        
        ⋅
        ]
        :
        R
        ⊗
        R
        →
        
          C
        
        [
        λ
        ]
        ⊗
        R
      
    
    {\displaystyle [\cdot _{\lambda }\cdot ]:R\otimes R\rightarrow \mathbb {C} [\lambda ]\otimes R}

<p>called the lambda bracket, which satisfies modified versions of bilinearity, skew-symmetry and the Jacobi identity:
</p>

[
        ∂
        
          a
          
            λ
          
        
        b
        ]
        =
        −
        λ
        [
        
          a
          
            λ
          
        
        b
        ]
        ,
        [
        
          a
          
            λ
          
        
        ∂
        b
        ]
        =
        (
        λ
        +
        ∂
        )
        [
        
          a
          
            λ
          
        
        b
        ]
        ,
      
    
    {\displaystyle [\partial a_{\lambda }b]=-\lambda [a_{\lambda }b],[a_{\lambda }\partial b]=(\lambda +\partial )[a_{\lambda }b],}

[
        
          a
          
            λ
          
        
        b
        ]
        =
        −
        [
        
          b
          
            −
            λ
            −
            ∂
          
        
        a
        ]
        ,
        
      
    
    {\displaystyle [a_{\lambda }b]=-[b_{-\lambda -\partial }a],\,}

[
        
          a
          
            λ
          
        
        [
        
          b
          
            μ
          
        
        c
        ]
        ]
        −
        [
        
          b
          
            μ
          
        
        [
        
          a
          
            λ
          
        
        c
        ]
        ]
        =
        [
        [
        
          a
          
            λ
          
        
        b
        
          ]
          
            λ
            +
            μ
          
        
        c
        ]
        .
        
      
    
    {\displaystyle [a_{\lambda }[b_{\mu }c]]-[b_{\mu }[a_{\lambda }c]]=[[a_{\lambda }b]_{\lambda +\mu }c].\,}

<p>One can see that removing all the lambda's, mu's and partials from the brackets, one simply has the definition of a Lie algebra.
</p>
<h2 id="examples-of-lie-conformal-algebras">Examples of Lie conformal algebras</h2>
<p>A simple and very important example of a Lie conformal algebra is the Virasoro conformal algebra. Over 
  
    
      
        
          C
        
        [
        ∂
        ]
      
    
    {\displaystyle \mathbb {C} [\partial ]}
  
 it is generated by a single element 
  
    
      
        L
      
    
    {\displaystyle L}
  
 with lambda bracket given by
</p>

[
        
          L
          
            λ
          
        
        L
        ]
        =
        (
        2
        λ
        +
        ∂
        )
        L
        .
        
      
    
    {\displaystyle [L_{\lambda }L]=(2\lambda +\partial )L.\,}

<p>In fact, it has been shown by Wakimoto that any Lie conformal algebra with lambda bracket satisfying the Jacobi identity on one generator is actually the Virasoro conformal algebra.
</p>
<h2 id="classification">Classification</h2>
<p>It has been shown that any finitely generated (as a 
  
    
      
        
          C
        
        [
        ∂
        ]
      
    
    {\displaystyle \mathbb {C} [\partial ]}
  
-module) simple Lie conformal algebra is isomorphic to either the Virasoro conformal algebra, a current conformal algebra or a semi-direct product of the two.
</p><p>There are also partial classifications of infinite subalgebras of 
  
    
      
        
          
            
              g
              c
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathfrak {gc}}_{n}}
  
 and 
  
    
      
        
          
            
              c
              e
              n
              d
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathfrak {cend}}_{n}}
  
.
</p>
<h2 id="generalizations">Generalizations</h2>

<h2 id="use-in-integrable-systems-and-relation-to-the-calculus-of-variations">Use in integrable systems and relation to the calculus of variations</h2>

<ul><li><a href="/facts/Victor_Kac/WdEv1WjL">Victor Kac</a>, "Vertex algebras for beginners". <i>University Lecture Series, 10.</i> American Mathematical Society, 1998. viii+141 pp. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-0643-2</li></ul>

Lie conformal algebra open-in-new

Lie conformal algebra