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 formula_1 a vector space isomorphism from the polynomial algebra formula_2 to the universal enveloping algebra formula_3. This map is not an algebra homomorphism. It is equivariant with respect to the natural representation of formula_1 on these spaces, so it restricts to a vector space  isomorphism where the superscript indicates the subspace annihilated by the action of formula_1. Both formula_7 and formula_8 are commutative subalgebras, indeed formula_8 is the center of formula_3, but formula_11 is still not an algebra homomorphism. However, Duflo proved that in some cases we can compose formula_11 with a map to get an algebra isomorphism Later, using using the Kontsevich formality theorem, Kontsevich showed that this works for all finite-dimensional Lie algebras. Following Calaque and Rossi, the map formula_15 can be defined as follows. The adjoint action of formula_1 is the map sending formula_18 to the operation formula_19 on formula_1 . We can treat map as an element of or, for that matter, an element of the larger space formula_22, since formula_23. Call this element Both formula_25 and formula_26 are algebras so their tensor product is as well. Thus, we can take powers of formula_27, say Going further, we can apply any formal power series to formula_27 and obtain an element of formula_30, where formula_31 denotes the algebra of formal power series on formula_32. Working with formal power series, we thus obtain an element Since the dimension of formula_34 is finite, one can think of formula_35 as formula_36, hence formula_37 is  formula_38 and by applying the determinant map, we obtain an element which is related to the Todd class in algebraic topology. Now, formula_32 acts as derivations on formula_2 since any element of formula_32 gives a translation-invariant vector field on formula_1. As a result, the algebra formula_44 acts on as differential operators on formula_2, and this extends to an action of formula_2 on formula_2. We can thus define a linear map by and since the whole construction was invariant, formula_15 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.