Inventiones mathematicae

, Volume 139, Issue 1, pp 135–172 | Cite as

The non-commutative Weil algebra

  • A. Alekseev
  • E. Meinrenken


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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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • A. Alekseev
    • 1
  • E. Meinrenken
    • 2
  1. 1.Institute for Theoretical Physics, Uppsala University, Box 803, S-75108 Uppsala, Sweden¶(e-mail:
  2. 2.University of Toronto, Department of Mathematics, 100 St George Street, Toronto, Ontario M5S3G3, Canada (e-mail:

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