Acta Mathematica Sinica, English Series

, Volume 24, Issue 2, pp 285–304 | Cite as

A superalgebraic interpretation of the quantization maps of Weil algebras



Let G be a Lie group whose Lie algebra \( \mathfrak{g} \) is quadratic. In the paper “the non-commutative Weil algebra”, Alekseev and Meinrenken constructed an explicit G-differential space homomorphism
, called the quantization map, between the Weil algebra \( W_\mathfrak{g} = S(\mathfrak{g}*) \otimes \wedge (\mathfrak{g}*) \) and
(which they call the noncommutative Weil algebra) for \( \mathfrak{g} \). They showed that
induces an algebra isomorphism between the basic cohomology rings H bas * (\( W_\mathfrak{g} \)) and H bas * (
). In this paper, we will interpret the quantization map
as the super Duflo map between the symmetric algebra \( S\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right) \) and the universal enveloping algebra \( U\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right) \) of a super Lie algebra \( \widetilde{T\mathfrak{g}\left[ 1 \right]} \) which is canonically associated with the quadratic Lie algebra \( \mathfrak{g} \). The basic cohomology rings H bas * (\( W_\mathfrak{g} \)) and H bas * (
) correspond exactly to \( S\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right)^{inv} \) and \( U\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right)^{inv} \), respectively. So what they proved is equivalent to the fact that the super Duflo map commutes with the adjoint action of the super Lie algebra, and that the super Duflo map is an algebra homomorphism when restricted to the space of invariants.


noncommutative Weil algebras quantization Duflo map G-differential algebras 

MR(2000) Subject Classification

46L65 17B70 17B63 81R25 


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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institute of Mathematical ScienceNanjing UniversityNanjingP. R. China

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