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 W G =S(?*)⊗∧?*, ? G carries the structure of an acyclic, locally free G-differential algebra and can be used to define equivariant cohomology ℋ G (B) for any G-differential algebra B. We construct an explicit isomorphism ?: W G →? 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 H G (B)→ℋ G (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.
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