In mathematics, the Duflo isomorphism is an isomorphism between the center of the universal enveloping algebra of a finite-dimensional Lie algebra over a field of characteristic zero and the invariants of its symmetric algebra. It was introduced by MichelDuflo(1977) and later generalized to arbitrary finite-dimensional Lie algebras by Maxim Kontsevich.

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

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

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

to get an algebra isomorphism

F ∘ G : S ( g ) g → U ( g ) g . {\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 G {\displaystyle G} can be defined as follows. The adjoint action of g {\displaystyle {\mathfrak {g}}} is the map

g → E n d ( g ) {\displaystyle {\mathfrak {g}}\to \mathrm {End} ({\mathfrak {g}})}

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

g ∗ ⊗ E n d ( g ) {\displaystyle {\mathfrak {g}}^{\ast }\otimes \mathrm {End} ({\mathfrak {g}})}

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

a d ∈ S ( g ∗ ) ⊗ E n d ( g ) {\displaystyle \mathrm {ad} \in S({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}

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

a d k ∈ S ( g ∗ ) ⊗ E n d ( g ) . {\displaystyle \mathrm {ad} ^{k}\in S({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}}).}

Going further, we can apply any formal power series to a d {\displaystyle \mathrm {ad} } and obtain an element of S ¯ ( g ∗ ) ⊗ E n d ( g ) {\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}, where S ¯ ( g ∗ ) {\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })} denotes the algebra of formal power series on g ∗ {\displaystyle {\mathfrak {g}}^{\ast }}. Working with formal power series and applying the hypothesis that g {\displaystyle {\mathfrak {g}}} is defined over a field of characteristic zero, we thus obtain an element

e a d / 2 − e − a d / 2 a d ∈ S ¯ ( g ∗ ) ⊗ E n d ( g ) {\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 g {\displaystyle {\mathfrak {g}}} is finite, one can think of E n d ( g ) {\displaystyle \mathrm {End} ({\mathfrak {g}})} as M n ( R ) {\displaystyle \mathrm {M} _{n}(\mathbb {R} )}, hence S ¯ ( g ∗ ) ⊗ E n d ( g ) {\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})} is M n ( S ¯ ( g ∗ ) ) {\displaystyle \mathrm {M} _{n}({\overline {S}}({\mathfrak {g}}^{\ast }))} and by applying the determinant map, we obtain an element

J ~ 1 / 2 := d e t e a d / 2 − e − a d / 2 a d ∈ S ¯ ( g ∗ ) {\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 })}

which is related to the Todd class in algebraic topology.

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

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

by

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

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

G : S ( g ) g → S ( g ) g . {\displaystyle 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.