The non-commutative Weil algebra

1999/03/09 by A. Alekseev, Anton Alekseev, Eckhard Meinrenken +1 · 139 citations
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Cellular algebra #Filtered algebra #Homotopy and Cohomology in Algebraic Topology #Mathematics #Pure mathematics #Ring (chemistry) #Subalgebra #Symmetric algebra #Universal enveloping algebra #math.DG

paper · pdf · doi:10.1007/s002229900025

published in Inventiones mathematicae 139(1), 135-172 (Springer Science+Business Media) · 34 pages

arxiv created 1999/03/09 · openalex publication_date 2000/01/01 · arxiv updated 2015/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo map extends to a linear map from compactly supported distributions on the Lie algebra g to compactly supported distributions on the Lie group G, which is a ring homomorphism for G-invariant distributions. In this paper we obtain analogues of the Duflo map and of Duflo's theorem in the context of equivariant cohomology of G-manifolds. Our result involves a non-commutative version of the Weil algebra and of the de Rham model of equivariant cohomology.

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