Duflo isomorphism

Template:No footnotes 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 Template:Harvs and later generalized to arbitrary finite-dimensional Lie algebras by Kontsevich.

The Poincaré-Birkoff-Witt theorem gives for any Lie algebra $𝔤$ a vector space isomorphism from the polynomial algebra $S(𝔤)$ to the universal enveloping algebra $U(𝔤)$. This map is not an algebra homomorphism. It is equivariant with respect to the natural representation of $𝔤$ on these spaces, so it restricts to a vector space isomorphism

$F:S(𝔤{)}^{𝔤}\to U(𝔤{)}^{𝔤}$

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

$G:S(𝔤{)}^{𝔤}\to S(𝔤{)}^{𝔤}$

to get an algebra isomorphism

$F\circ G:S(𝔤{)}^{𝔤}\to U(𝔤{)}^{𝔤}.$

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 $𝔤$ is the map

$𝔤\to \mathrm{E}\mathrm{n}\mathrm{d}(𝔤)$

sending $x\in 𝔤$ to the operation $[x,-]$ on $𝔤$. We can treat map as an element of

${𝔤}^{\ast }\otimes \mathrm{E}\mathrm{n}\mathrm{d}(𝔤)$

or, for that matter, an element of the larger space $S({𝔤}^{\ast })\otimes \mathrm{E}\mathrm{n}\mathrm{d}(𝔤)$, since ${𝔤}^{\ast }\subset S({𝔤}^{\ast })$. Call this element

$\mathrm{a}\mathrm{d}\in S({𝔤}^{\ast })\otimes \mathrm{E}\mathrm{n}\mathrm{d}(𝔤)$

Both $S({𝔤}^{\ast })$ and $\mathrm{E}\mathrm{n}\mathrm{d}(𝔤)$ are algebras so their tensor product is as well. Thus, we can take powers of $\mathrm{a}\mathrm{d}$, say

${\mathrm{a}\mathrm{d}}^k\in S({𝔤}^{\ast })\otimes \mathrm{E}\mathrm{n}\mathrm{d}(𝔤).$

Going further, we can apply any formal power series to $\mathrm{a}\mathrm{d}$ and obtain an element of $\bar{S}({𝔤}^{\ast })\otimes \mathrm{E}\mathrm{n}\mathrm{d}(𝔤)$, where $\bar{S}({𝔤}^{\ast })$ denotes the algebra of formal power series on ${𝔤}^{\ast }$. Working with formal power series, we thus obtain an element

$\sqrt{\frac{e^{\mathrm{a}\mathrm{d}}-e^{-\mathrm{a}\mathrm{d}}}{\mathrm{a}\mathrm{d}}}\in \bar{S}({𝔤}^{\ast })\otimes \mathrm{E}\mathrm{n}\mathrm{d}(𝔤)$

Since the dimension of $𝔤$ is finite, one can think of $\mathrm{E}\mathrm{n}\mathrm{d}(𝔤)$ as ${\mathrm{M}}_n(ℝ)$, hence $\bar{S}({𝔤}^{\ast })\otimes \mathrm{E}\mathrm{n}\mathrm{d}(𝔤)$ is ${\mathrm{M}}_n(\bar{S}({𝔤}^{\ast }))$ and by applying the determinant map, we obtain an element

${\tilde{J}}^{1/2}:=\mathrm{d}\mathrm{e}\mathrm{t}\sqrt{\frac{e^{\mathrm{a}\mathrm{d}}-e^{-\mathrm{a}\mathrm{d}}}{\mathrm{a}\mathrm{d}}}\in \bar{S}({𝔤}^{\ast })$

which is related to the Todd class in algebraic topology.

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

$G:S(𝔤)\to S(𝔤)$

by

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

and since the whole construction was invariant, $G$ restricts to the desired linear map

$G:S(𝔤{)}^{𝔤}\to S(𝔤{)}^{𝔤}.$

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.

References

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