Duflo isomorphism

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

[math]\displaystyle{ F\colon S(\mathfrak{g})^{\mathfrak{g}} \to U(\mathfrak{g})^{\mathfrak{g}} }[/math]

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

[math]\displaystyle{ G \colon S(\mathfrak{g})^{\mathfrak{g}} \to S(\mathfrak{g})^{\mathfrak{g}} }[/math]

to get an algebra isomorphism

[math]\displaystyle{ F \circ G \colon S(\mathfrak{g})^{\mathfrak{g}} \to U(\mathfrak{g})^{\mathfrak{g}} . }[/math]

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

Following Calaque and Rossi, the map [math]\displaystyle{ G }[/math] can be defined as follows. The adjoint action of [math]\displaystyle{ \mathfrak{g} }[/math] is the map

[math]\displaystyle{ \mathfrak{g} \to \mathrm{End}(\mathfrak{g}) }[/math]

sending [math]\displaystyle{ x \in \mathfrak{g} }[/math] to the operation [math]\displaystyle{ [x,-] }[/math] on [math]\displaystyle{ \mathfrak{g} }[/math]. We can treat map as an element of

[math]\displaystyle{ \mathfrak{g}^\ast \otimes \mathrm{End}(\mathfrak{g}) }[/math]

or, for that matter, an element of the larger space [math]\displaystyle{ S(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g}) }[/math], since [math]\displaystyle{ \mathfrak{g}^\ast \subset S(\mathfrak{g}^\ast) }[/math]. Call this element

[math]\displaystyle{ \mathrm{ad} \in S(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g}) }[/math]

Both [math]\displaystyle{ S(\mathfrak{g}^\ast) }[/math] and [math]\displaystyle{ \mathrm{End}(\mathfrak{g}) }[/math] are algebras so their tensor product is as well. Thus, we can take powers of [math]\displaystyle{ \mathrm{ad} }[/math], say

[math]\displaystyle{ \mathrm{ad}^k \in S(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g}). }[/math]

Going further, we can apply any formal power series to [math]\displaystyle{ \mathrm{ad} }[/math] and obtain an element of [math]\displaystyle{ \overline{S}(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g}) }[/math], where [math]\displaystyle{ \overline{S}(\mathfrak{g}^\ast) }[/math] denotes the algebra of formal power series on [math]\displaystyle{ \mathfrak{g}^\ast }[/math]. Working with formal power series, we thus obtain an element

[math]\displaystyle{ \sqrt{\frac{e^{\mathrm{ad}} - e^{-\mathrm{ad}}}{\mathrm{ad}}} \in \overline{S}(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g}) }[/math]

Since the dimension of [math]\displaystyle{ \mathfrak{g} }[/math] is finite, one can think of [math]\displaystyle{ \mathrm{End}(\mathfrak{g}) }[/math] as [math]\displaystyle{ \mathrm{M}_n(\mathbb{R}) }[/math], hence [math]\displaystyle{ \overline{S}(\mathfrak{g}^\ast) \otimes \mathrm{End}(\mathfrak{g}) }[/math] is [math]\displaystyle{ \mathrm{M}_n(\overline{S}(\mathfrak{g}^\ast)) }[/math] and by applying the determinant map, we obtain an element

[math]\displaystyle{ \tilde{J}^{1/2} := \mathrm{det} \sqrt{\frac{e^{\mathrm{ad}} - e^{-\mathrm{ad}}}{\mathrm{ad}}} \in \overline{S}(\mathfrak{g}^\ast) }[/math]

which is related to the Todd class in algebraic topology.

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

[math]\displaystyle{ G \colon S(\mathfrak{g}) \to S(\mathfrak{g}) }[/math]

by

[math]\displaystyle{ G(\psi) = \tilde{J}^{1/2} \psi }[/math]

and since the whole construction was invariant, [math]\displaystyle{ G }[/math] restricts to the desired linear map

[math]\displaystyle{ G \colon S(\mathfrak{g})^{\mathfrak{g}} \to S(\mathfrak{g})^{\mathfrak{g}} . }[/math]


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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