In mathematics, the Duflo isomorphism is an isomorphism between the center of the universal enveloping algebra of a finite-dimensional Lie algebra over a field of characteristic zero and the invariants of its symmetric algebra. It was introduced by and later generalized to arbitrary finite-dimensional Lie algebras by Maxim 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 and applying the hypothesis that is defined over a field of characteristic zero, 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 Calaque 2011, p. 4 (Remark 1.4). 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.
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