The non-commutative Weil algebra |
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Authors: | A Alekseev E Meinrenken |
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Institution: | Institute for Theoretical Physics, Uppsala University, Box 803, S-75108 Uppsala, Sweden?(e-mail: alekseev@teorfys.uu.se), SE University of Toronto, Department of Mathematics, 100 St George Street, Toronto, Ontario M5S3G3, Canada (e-mail: mein@math.toronto.edu), CA
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Abstract: | 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.
Oblatum 13-III-1999 & 27-V-1999 / Published online: 22 September 1999 |
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Keywords: | |
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