Duflo isomorphism

rdf:langString Duflo isomorphism

rdf:langString Michel

rdf:langString Duflo

rdf:langString 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 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 . Both and are commutative subalgebras, indeed is the center of , but is still not an algebra homomorphism. However, Duflo proved that in some cases we can compose with a map to get an algebra isomorphism Later, using the , 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 is the map sending to the operation on . We can treat map as an element of or, for that matter, an element of the larger space , since . Call this element Both and are algebras so their tensor product is as well. Thus, we can take powers of , say Going further, we can apply any formal power series to and obtain an element of , where denotes the algebra of formal power series on . Working with formal power series, we thus obtain an element Since the dimension of is finite, one can think of as , hence is and by applying the determinant map, we obtain an element which is related to the Todd class in algebraic topology. Now, acts as derivations on since any element of gives a translation-invariant vector field on . As a result, the algebra acts on as differential operators on , and this extends to an action of on . We can thus define a linear map by and since the whole construction was invariant, restricts to the desired linear map

rdf:langString Michel Duflo