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 $\mathfrak$ a vector space isomorphism from the polynomial algebra $S(\mathfrak)$ to the universal enveloping algebra $U(\mathfrak)$. This map is not an algebra homomorphism. It is equivariant with respect to the natural representation of $\mathfrak$ on these spaces, so it restricts to a vector space isomorphism
: $F\colon S(\mathfrak)^ \to U(\mathfrak)^$
where the superscript indicates the subspace annihilated by the action of $\mathfrak$. Both $S(\mathfrak)^$ and $U(\mathfrak)^$ are commutative subalgebras, indeed $U(\mathfrak)^$ is the center of $U(\mathfrak)$, but $F$ is still not an algebra homomorphism. However, Duflo proved that in some cases we can compose $F$ with a map
: $G \colon S(\mathfrak)^ \to S(\mathfrak)^$
to get an algebra isomorphism
:$F \circ G \colon S(\mathfrak)^ \to U(\mathfrak)^ .$
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 $\mathfrak$ is the map
: $\mathfrak \to \mathrm(\mathfrak)$
sending $x \in \mathfrak$ to the operation ,-/math> on $\mathfrak$. We can treat map as an element of
: $\mathfrak^\ast \otimes \mathrm(\mathfrak)$
or, for that matter, an element of the larger space $S(\mathfrak^\ast) \otimes \mathrm(\mathfrak)$, since $\mathfrak^\ast \subset S(\mathfrak^\ast)$. Call this element
: $\mathrm \in S(\mathfrak^\ast) \otimes \mathrm(\mathfrak)$
Both $S(\mathfrak^\ast)$ and $\mathrm(\mathfrak)$ are algebras so their tensor product is as well. Thus, we can take powers of $\mathrm$, say
: $\mathrm^k \in S(\mathfrak^\ast) \otimes \mathrm(\mathfrak).$
Going further, we can apply any formal power series to $\mathrm$ and obtain an element of $\overline(\mathfrak^\ast) \otimes \mathrm(\mathfrak)$, where $\overline(\mathfrak^\ast)$ denotes the algebra of formal power series on $\mathfrak^\ast$. Working with formal power series, we thus obtain an element
: $\sqrt \in \overline(\mathfrak^\ast) \otimes \mathrm(\mathfrak)$
Since the dimension of $\mathfrak$ is finite, one can think of $\mathrm(\mathfrak)$ as $\mathrm_n(\mathbb)$, hence $\overline(\mathfrak^\ast) \otimes \mathrm(\mathfrak)$ is $\mathrm_n(\overline(\mathfrak^\ast))$ and by applying the determinant map, we obtain an element
: $\tilde^ := \mathrm \sqrt \in \overline(\mathfrak^\ast)$
which is related to the Todd class in algebraic topology.

Now, $\mathfrak^\ast$ acts as derivations on $S(\mathfrak)$ since any element of $\mathfrak^\ast$ gives a translation-invariant vector field on $\mathfrak$. As a result, the algebra $S(\mathfrak^\ast)$ acts on
as differential operators on $S(\mathfrak)$, and this extends to an action of $\overline(\mathfrak^\ast)$ on $S(\mathfrak)$. We can thus define a linear map
: $G \colon S(\mathfrak) \to S(\mathfrak)$
by
: $G(\psi) = \tilde^ \psi$
and since the whole construction was invariant, $G$ restricts to the desired linear map
: $G \colon S(\mathfrak)^ \to S(\mathfrak)^ .$

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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Lie algebras

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