The non-commutative Weil algebra

For any compact Lie group G, together with an invariant inner product on its Lie algebra ?, we define the non-commutative Weil algebra ?G as a tensor product of the universal enveloping algebra U(?) and the Clifford algebra Cl(?). Just like the usual Weil algebra WG=S(?*)⊗∧?*, ?G carries the structure of an acyclic, locally free G-differential algebra and can be used to define equivariant cohomology ℋG(B) for any G-differential algebra B. We construct an explicit isomorphism ?: WG→?G of the two Weil algebras as G-differential spaces, and prove that their multiplication maps are G-chain homotopic. This implies that the map in cohomology HG(B)→ℋG(B) induced by ? is a ring isomorphism. For the trivial G-differential algebra B=ℝ, this reduces to the Duflo isomorphism S(?)G≅U(?)G between the ring of invariant polynomials and the ring of Casimir elements.