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
a vector space isomorphism from the polynomial algebra
to the universal enveloping algebra
This map is not an algebra homomorphism.
It is equivariant with respect to the natural representation of
on these spaces, so it restricts to a vector space isomorphism where the superscript indicates the subspace annihilated by the action of
are commutative subalgebras, indeed
However, Duflo proved that in some cases we can compose
with a map to get an algebra isomorphism Later, using the Kontsevich formality theorem, Kontsevich showed that this works for all finite-dimensional Lie algebras.
Following Calaque and Rossi, the map
The adjoint action of
is the map sending
We can treat map as an element of or, for that matter, an element of the larger space
Call this element Both
are algebras so their tensor product is as well.
, say Going further, we can apply any formal power series to
denotes the algebra of formal power series on
Working with formal power series, we thus obtain an element Since the dimension of
and by applying the determinant map, we obtain an element which is related to the Todd class in algebraic topology.
acts as derivations on
gives a translation-invariant vector field on
acts on as differential operators on
, and this extends to an action of
We can thus define a linear map by and since the whole construction was invariant,
restricts to the desired linear map
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.
This abstract algebra-related article is a stub.