Duflo isomorphism explained

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

ak{g}

a vector space isomorphism from the polynomial algebra

S(ak{g})

to the universal enveloping algebra

U(ak{g})

. This map is not an algebra homomorphism. It is equivariant with respect to the natural representation of

ak{g}

on these spaces, so it restricts to a vector space isomorphism

F\colonS(ak{g})^ak{g

} \to U(\mathfrak)^ where the superscript indicates the subspace annihilated by the action of

ak{g}

. Both

S(ak{g})^ak{g

} and

U(ak{g})^ak{g

} are commutative subalgebras, indeed

U(ak{g})^ak{g

} is the center of

U(ak{g})

, but

is still not an algebra homomorphism. However, Duflo proved that in some cases we can compose

with a map

G\colonS(ak{g})^ak{g

} \to S(\mathfrak)^ to get an algebra isomorphism

F\circG\colonS(ak{g})^ak{g

} \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

can be defined as follows. The adjoint action of

ak{g}

is the map

ak{g}\toEnd(ak{g})

sending

x\inak{g}

to the operation

[x,-]

ak{g}

. We can treat map as an element of

ak{g}^\ast ⊗ End(ak{g})

or, for that matter, an element of the larger space

S(ak{g}^\ast) ⊗ End(ak{g})

, since

ak{g}^\ast\subsetS(ak{g}^\ast)

. Call this element

ad\inS(ak{g}^\ast) ⊗ End(ak{g})

Both

S(ak{g}^\ast)

and

End(ak{g})

are algebras so their tensor product is as well. Thus, we can take powers of

, say

ad^k\inS(ak{g}^\ast) ⊗ End(ak{g}).

Going further, we can apply any formal power series to

and obtain an element of

\overline{S}(ak{g}^\ast) ⊗ End(ak{g})

, where

\overline{S}(ak{g}^\ast)

denotes the algebra of formal power series on

ak{g}^\ast

. Working with formal power series, we thus obtain an element

\sqrt{	e^ad-e^-ad
	ad

} \in \overline(\mathfrak^\ast) \otimes \mathrm(\mathfrak) Since the dimension of

ak{g}

is finite, one can think of

End(ak{g})

M_n(R)

, hence

\overline{S}(ak{g}^\ast) ⊗ End(ak{g})

	\ast))
M
	n(\overline{S}(ak{g}

and by applying the determinant map, we obtain an element

\tilde{J}^1/2:=det\sqrt{

	e^ad-e^-ad
	ad

} \in \overline(\mathfrak^\ast) which is related to the Todd class in algebraic topology.

Now,

ak{g}^\ast

acts as derivations on

S(ak{g})

since any element of

ak{g}^\ast

gives a translation-invariant vector field on

ak{g}

. As a result, the algebra

S(ak{g}^\ast)

acts on as differential operators on

S(ak{g})

, and this extends to an action of

\overline{S}(ak{g}^\ast)

S(ak{g})

. We can thus define a linear map

G\colonS(ak{g})\toS(ak{g})

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

and since the whole construction was invariant,

restricts to the desired linear map

G\colonS(ak{g})^ak{g

} \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.