International Congress of Mathematicians

Clifford Algebras and the Duflo Isomorphism 641Theorem. [1] The quantization mapQ := (sym®q) o i(Si) : Wg -• Wgintertwines the contraction operators, Lie derivatives, and differentials on Wg andon Wg.The fact that Q intertwines the two differentials d 14 , d w relies on a numberof special properties of the function S e , including the CDYBE.Put differently, the quantization map Q defines a new, graded non-commutativering structure on the Weil algebra Wg, in such a way that the derivations i)fi, Lf , dare still derivations for the new ring structure, and in fact become inner derivations.Notice that Q restricts to the quantization map for Clifford algebras q : Ag —¥ Cl(g)on the second factor and to the Duflo map on the first factor, but is not just theproduct of these two maps.5. Equivariant cohomologyH. Cartan in [3] introduced the Weil algebra Wg as an algebraic model for thealgebra of differential forms on the classifying bundle EG, at least in the case Gcompact.In particular, it can be used to compute the equivariant cohomology HQ(M)(with real coefficients) for any G-manifold M. Yet if h ,Lf h ,d Rh denote the contractionoperators, Lie derivatives, and differential on the de Rham complex 0(Af )of differential forms. LetH B (M) = H((Wg ® tt(M)) basic , d w + d m )where (Wg ® 0(Af ))t> as ic is the subspace annihilated by all Lie derivatives Lf' +L Rh and all contraction operators if' + i Rh . Cartan's result says that H S (M) =HQ(M,R) provided G is compact.Alore generally, we can define H B (A) for any g-differential algebra A. YetR S (A) be defined by replacing Wg with Wg. The quantization map Q : Wg —¥ Wginduces a map Q : H S (A) —¥ R S (A).Theorem. [1] For any g-differential algebra A, the vector space isomorphism Q :H s (Â) —¥ R s (A) is in fact an algebra isomorphism.Our proof is by construction of an explicit chain homotopy between the twomaps Wg ® Wg —¥ Wg given by "quantization followed by multiplication" and"multiplication followed by quantization", respectively. Taking A to be the trivial g-differential algebra (i.e. A = Q(point)), the statement specializes to Duflo's theoremfor quadratic g.References[1] A. Alekseev & E. Aleinrenken, The non-commutative Weil algebra, Invent.Math. 139 (2000), 135^172.