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 $$\mathfrak{g}$$ a vector space isomorphism from the polynomial algebra $$S(\mathfrak{g})$$ to the universal enveloping algebra $$U(\mathfrak{g})$$. This map is not an algebra homomorphism. It is equivariant with respect to the natural representation of $$\mathfrak{g}$$ on these spaces, so it restricts to a vector space isomorphism
 * $$F\colon S(\mathfrak{g})^{\mathfrak{g}} \to U(\mathfrak{g})^{\mathfrak{g}} $$

where the superscript indicates the subspace annihilated by the action of $$\mathfrak{g}$$. Both $$S(\mathfrak{g})^{\mathfrak{g}}$$ and $$U(\mathfrak{g})^{\mathfrak{g}}$$ are commutative subalgebras, indeed $$U(\mathfrak{g})^{\mathfrak{g}}$$ is the center of $$U(\mathfrak{g})$$, but $$F$$ is still not an algebra homomorphism. However, Duflo proved that in some cases we can compose $$F$$ with a map
 * $$G \colon S(\mathfrak{g})^{\mathfrak{g}} \to S(\mathfrak{g})^{\mathfrak{g}} $$

to get an algebra isomorphism
 * $$ F \circ G \colon S(\mathfrak{g})^{\mathfrak{g}} \to  U(\mathfrak{g})^{\mathfrak{g}} .$$

Later, using the Kontsevich formality theorem, Kontsevich showed that this works for all finite-dimensional Lie algebras.

Following Calaque and Rossi, the map $$G$$ can be defined as follows. The adjoint action of $$\mathfrak{g}$$ is the map
 * $$ \mathfrak{g} \to \mathrm{End}(\mathfrak{g}) $$

sending $$x \in \mathfrak{g}$$ to the operation $$[x,-]$$ on $$\mathfrak{g}$$. We can treat map as an element of
 * $$ \mathfrak{g}^\ast \otimes \mathrm{End}(\mathfrak{g})$$

or, for that matter, an element of the larger space $$S(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g})$$, since $$ \mathfrak{g}^\ast \subset S(\mathfrak{g}^\ast)$$. Call this element
 * $$ \mathrm{ad} \in S(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g})$$

Both $$S(\mathfrak{g}^\ast)$$ and $$\mathrm{End}(\mathfrak{g})$$ are algebras so their tensor product is as well. Thus, we can take powers of $$\mathrm{ad}$$, say
 * $$ \mathrm{ad}^k \in S(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g}).$$

Going further, we can apply any formal power series to $$\mathrm{ad}$$ and obtain an element of $$\overline{S}(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g})$$, where $$\overline{S}(\mathfrak{g}^\ast)$$ denotes the algebra of formal power series on $$\mathfrak{g}^\ast$$. Working with formal power series, we thus obtain an element
 * $$ \sqrt{\frac{e^{\mathrm{ad}} - e^{-\mathrm{ad}}}{\mathrm{ad}}} \in \overline{S}(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g}) $$

Since the dimension of $$ \mathfrak{g} $$ is finite, one can think of $$ \mathrm{End}(\mathfrak{g})$$ as $$ \mathrm{M}_n(\mathbb{R})$$, hence $$ \overline{S}(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g}) $$ is $$ \mathrm{M}_n(\overline{S}(\mathfrak{g}^\ast)) $$ and by applying the determinant map, we obtain an element
 * $$ \tilde{J}^{1/2} := \mathrm{det} \sqrt{\frac{e^{\mathrm{ad}} - e^{-\mathrm{ad}}}{\mathrm{ad}}} \in \overline{S}(\mathfrak{g}^\ast) $$

which is related to the Todd class in algebraic topology.

Now, $$\mathfrak{g}^\ast$$ acts as derivations on $$S(\mathfrak{g})$$ since any element of $$\mathfrak{g}^\ast$$ gives a translation-invariant vector field on $$\mathfrak{g}$$. As a result, the algebra $$S(\mathfrak{g}^\ast) $$ acts on as differential operators on $$S(\mathfrak{g})$$, and this extends to an action of $$\overline{S}(\mathfrak{g}^\ast)$$ on $$S(\mathfrak{g})$$. We can thus define a linear map
 * $$G \colon S(\mathfrak{g}) \to S(\mathfrak{g}) $$

by
 * $$ G(\psi) = \tilde{J}^{1/2} \psi $$

and since the whole construction was invariant, $$G$$ restricts to the desired linear map
 * $$G \colon S(\mathfrak{g})^{\mathfrak{g}} \to S(\mathfrak{g})^{\mathfrak{g}} .$$

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.