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

${\displaystyle F\colon S({\mathfrak {g}})^{\mathfrak {g}}\to U({\mathfrak {g}})^{\mathfrak {g}}}$

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

${\displaystyle G\colon S({\mathfrak {g}})^{\mathfrak {g}}\to S({\mathfrak {g}})^{\mathfrak {g}}}$

to get an algebra isomorphism

${\displaystyle F\circ G\colon S({\mathfrak {g}})^{\mathfrak {g}}\to U({\mathfrak {g}})^{\mathfrak {g}}.}$

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

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

${\displaystyle {\mathfrak {g}}\to \mathrm {End} ({\mathfrak {g}})}$

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

${\displaystyle {\mathfrak {g}}^{\ast }\otimes \mathrm {End} ({\mathfrak {g}})}$

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

${\displaystyle \mathrm {ad} \in S({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}$

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

${\displaystyle \mathrm {ad} ^{k}\in S({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}}).}$

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

${\displaystyle {\sqrt {\frac {e^{\mathrm {ad} /2}-e^{-\mathrm {ad} /2}}{\mathrm {ad} }}}\in {\overline {S}}({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}$

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

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

which is related to the Todd class in algebraic topology.

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

${\displaystyle G\colon S({\mathfrak {g}})\to S({\mathfrak {g}})}$

by

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

and since the whole construction was invariant, ${\displaystyle G}$ restricts to the desired linear map

${\displaystyle G\colon S({\mathfrak {g}})^{\mathfrak {g}}\to S({\mathfrak {g}})^{\mathfrak {g}}.}$

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.

• Duflo, Michel (1977), "Opérateurs différentiels bi-invariants sur un groupe de Lie", Annales Scientifiques de l'École Normale Supérieure, Série 4, 10 (2): 265–288, doi:10.24033/asens.1327, ISSN 0012-9593, MR 0444841
• Calaque, Damien; Rossi, Carlo A. (2011), Lectures on Duflo isomorphisms in Lie algebra and complex geometry, EMS Series of Lectures in Mathematics, Zürich: European Mathematical Society, doi:10.4171/096, hdl:21.11116/0000-0004-2127-B, ISBN 978-3-03719-096-8, MR 2816610