Duflo isomorphism
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  and  later generalized to arbitrary finite-dimensional Lie algebras by Kontsevich. The Poincaré-Birkoff-Witt theorem gives for any Lie algebra  a vector space isomorphism from the polynomial algebra  to the universal enveloping algebra . This map is not an algebra homomorphism. It is equivariant with respect to the natural representation of  on these spaces, so it restricts to a vector space isomorphism
 
where the superscript indicates the subspace annihilated by the action of . Both  and  are commutative subalgebras, indeed  is the center of , but  is still not an algebra homomorphism. However, Duflo proved that in some cases we can compose  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  can be defined as follows. The adjoint action of  is the map
 
sending  to the operation  on . We can treat map as an element of
 
or, for that matter, an element of the larger space , since . Call this element
 
Both  and  are algebras so their tensor product is as well. Thus, we can take powers of , say
 
Going further, we can apply any formal power series to  and obtain an element of , where  denotes the algebra of formal power series on . Working with formal power series, we thus obtain an element
 
Since the dimension of  is finite, one can think of  as , hence  is  and by applying the determinant map, we obtain an element
 
which is related to the Todd class in algebraic topology. Now,  acts as derivations on  since any element of  gives a translation-invariant vector field on . As a result, the algebra  acts on 
as differential operators on , and this extends to an action of  on . We can thus define a linear map
 
by
 
and since the whole construction was invariant,  restricts to the desired linear map

Properties
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. References

Lie algebras