In mathematics, the Duflo isomorphism is an isomorphism between the center of the universal enveloping algebra of a finite-dimensional Lie algebra and the invariants of its symmetric algebra. It was introduced by Michel Duflo (1977) and  later generalized to arbitrary finite-dimensional Lie algebras by Kontsevich.
 The Poincaré-Birkoff-Witt theorem gives for any Lie algebra 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
  
 a vector space isomorphism from the polynomial algebra 
  
    
      
        S
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle S({\mathfrak {g}})}
  
 to the universal enveloping algebra 
  
    
      
        U
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle U({\mathfrak {g}})}
  
.   This map is not an algebra homomorphism.    It is equivariant with respect to the natural representation of 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
  
 on these spaces, so it restricts to a vector space isomorphism
 where the superscript indicates the subspace annihilated by the action of 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
  
.   Both 
  
    
      
        S
        (
        
          
            g
          
        
        
          )
          
            
              g
            
          
        
      
    
    {\displaystyle S({\mathfrak {g}})^{\mathfrak {g}}}
  
 and 
  
    
      
        U
        (
        
          
            g
          
        
        
          )
          
            
              g
            
          
        
      
    
    {\displaystyle U({\mathfrak {g}})^{\mathfrak {g}}}
  
 are commutative subalgebras, indeed 
  
    
      
        U
        (
        
          
            g
          
        
        
          )
          
            
              g
            
          
        
      
    
    {\displaystyle U({\mathfrak {g}})^{\mathfrak {g}}}
  
 is the center of 
  
    
      
        U
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle U({\mathfrak {g}})}
  
, but 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is still not an algebra homomorphism.  However, Duflo proved that in some cases we can compose 
  
    
      
        F
      
    
    {\displaystyle F}
  
 with a map 
 to get an algebra isomorphism
 Later, using the Kontsevich formality theorem, Kontsevich showed that this works for all finite-dimensional Lie algebras.  
 Following Calaque and Rossi, the map 
  
    
      
        G
      
    
    {\displaystyle G}
  
 can be defined as follows.  The adjoint action of 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
  
 is the map
 sending 
  
    
      
        x
        ∈
        
          
            g
          
        
      
    
    {\displaystyle x\in {\mathfrak {g}}}
  
 to the operation 
  
    
      
        [
        x
        ,
        −
        ]
      
    
    {\displaystyle [x,-]}
  
 on 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
  
.   We can treat map as an element of
 or, for that matter, an element of the larger space 
  
    
      
        S
        (
        
          
            
              g
            
          
          
            ∗
          
        
        )
        ⊗
        
          E
          n
          d
        
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle S({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}
  
, since 
  
    
      
        
          
            
              g
            
          
          
            ∗
          
        
        ⊂
        S
        (
        
          
            
              g
            
          
          
            ∗
          
        
        )
      
    
    {\displaystyle {\mathfrak {g}}^{\ast }\subset S({\mathfrak {g}}^{\ast })}
  
.  Call this element
 Both 
  
    
      
        S
        (
        
          
            
              g
            
          
          
            ∗
          
        
        )
      
    
    {\displaystyle S({\mathfrak {g}}^{\ast })}
  
 and 
  
    
      
        
          E
          n
          d
        
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle \mathrm {End} ({\mathfrak {g}})}
  
 are algebras so their tensor product is as well.  Thus, we can take powers of 
  
    
      
        
          a
          d
        
      
    
    {\displaystyle \mathrm {ad} }
  
, say
 Going further, we can apply any formal power series to 
  
    
      
        
          a
          d
        
      
    
    {\displaystyle \mathrm {ad} }
  
 and obtain an element of 
  
    
      
        
          
            S
            ¯
          
        
        (
        
          
            
              g
            
          
          
            ∗
          
        
        )
        ⊗
        
          E
          n
          d
        
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}
  
, where 
  
    
      
        
          
            S
            ¯
          
        
        (
        
          
            
              g
            
          
          
            ∗
          
        
        )
      
    
    {\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })}
  
 denotes the algebra of formal power series on 
  
    
      
        
          
            
              g
            
          
          
            ∗
          
        
      
    
    {\displaystyle {\mathfrak {g}}^{\ast }}
  
.   Working with formal power series, we thus obtain an element
 Since the dimension of 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
  
 is finite, one can think of 
  
    
      
        
          E
          n
          d
        
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle \mathrm {End} ({\mathfrak {g}})}
  
 as 
  
    
      
        
          
            M
          
          
            n
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle \mathrm {M} _{n}(\mathbb {R} )}
  
, hence 
  
    
      
        
          
            S
            ¯
          
        
        (
        
          
            
              g
            
          
          
            ∗
          
        
        )
        ⊗
        
          E
          n
          d
        
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}
  
 is 
  
    
      
        
          
            M
          
          
            n
          
        
        (
        
          
            S
            ¯
          
        
        (
        
          
            
              g
            
          
          
            ∗
          
        
        )
        )
      
    
    {\displaystyle \mathrm {M} _{n}({\overline {S}}({\mathfrak {g}}^{\ast }))}
  
 and by applying the determinant map, we obtain an element
 which is related to the Todd class in algebraic topology.
 Now, 
  
    
      
        
          
            
              g
            
          
          
            ∗
          
        
      
    
    {\displaystyle {\mathfrak {g}}^{\ast }}
  
 acts as derivations on 
  
    
      
        S
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle S({\mathfrak {g}})}
  
 since any element of 
  
    
      
        
          
            
              g
            
          
          
            ∗
          
        
      
    
    {\displaystyle {\mathfrak {g}}^{\ast }}
  
 gives a translation-invariant vector field on 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
  
.   As a result, the algebra 
  
    
      
        S
        (
        
          
            
              g
            
          
          
            ∗
          
        
        )
      
    
    {\displaystyle S({\mathfrak {g}}^{\ast })}
  
 acts on 
as differential operators on 
  
    
      
        S
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle S({\mathfrak {g}})}
  
, and this extends to an action of 
  
    
      
        
          
            S
            ¯
          
        
        (
        
          
            
              g
            
          
          
            ∗
          
        
        )
      
    
    {\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })}
  
 on 
  
    
      
        S
        (
        
          
            g
          
        
        )
      
    
    {\displaystyle S({\mathfrak {g}})}
  
.   We can thus define a linear map
 by
 and since the whole construction was invariant, 
  
    
      
        G
      
    
    {\displaystyle G}
  
 restricts to the desired linear map
  
 For a nilpotent Lie algebra the Duflo isomorphism coincides with the symmetrization map from symmetric algebra to universal enveloping algebra. For a semisimple Lie algebra the Duflo isomorphism is  compatible in a natural way with the Harish-Chandra isomorphism.
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