# 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 Michel Duflo (1977) 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.