The Duflo map is the map S(g) -> U(g), which known to satisfy the following properties:

1) identity on g

2) isomorphism of g-modules (and in particular vector spaces)

3) restricted to Poisson center on S(g) it is ISOMORPHISM of commutative algebras S(g)^g to ZU(g) (the center of U(g)). (This is highly non-trivial property). It predicts that the centers on the "classical" and "quantum" level are the same. (Kontesevich generalized to arbitrary Poisson variety).

The question: Is it the only map satisfying such properties ? (At least for semisimple Lie algebras) ?

(I think answer is YES, and it should be known, but I have not seen the reference).

========= Example let g-commutative, then it is true:

Since S(g)=U(g) and since the map is required to be identity on g and it is homomorphism, so it is identity map on the S(g).

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If you read this question I guess you know what S(g) and U(g) means :) But may be nevertheless to keep good spirit of MO it is more polite to include definitions:

S(g) - means commutative algebra - symmetric algebra of g (i.e. just take some basis xi in g and consider polynomial algebra C[x1 ... xn] - this is S(g) as a vector space. Lie algebra g acts on - it simple way - it acts on xi by adjoint action, and continued further by Leibniz rule).

U(g) - means NON-commutative algebra - universal enveloping algebra of g - which is: you take basis xi and consider non-commutative polynomials C[xi] where generators satisfying the relations: [xi, xj] = C_ij^k x_k , where C_ij^k - are structure constants of g.

The Duflo map is for example discussed here:

D. Calaque, C. Rossi "Lectures on Duflo isomorphisms in Lie algebras and complex geometry"

http://people.mpim-bonn.mpg.de/crossi/LectETHbook.pdf

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Kontsevich mentions that in general situation (of Poisson manifolds) Grothendieck-Teichmuller groups should act on quantizations and in particular on "Duflo" type maps, but he writes that for semisimple algebras this action is trivial. So we should not have problem from this side. Moreover as far as I understand since nobody seen non-trivial example of this action it might be always like this.