Let G be a Lie group whose Lie algebra \documentclass[12pt]{minimal}
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\mathfrak{g}
$$\end{document} is quadratic. In the paper “the non-commutative Weil algebra”, Alekseev and Meinrenken constructed an explicit G-differential space homomorphism [graphic not available: see fulltext], called the quantization map, between the Weil algebra \documentclass[12pt]{minimal}
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W_\mathfrak{g} = S(\mathfrak{g}*) \otimes \wedge (\mathfrak{g}*)
$$\end{document} and [graphic not available: see fulltext] (which they call the noncommutative Weil algebra) for \documentclass[12pt]{minimal}
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\mathfrak{g}
$$\end{document}. They showed that [graphic not available: see fulltext] induces an algebra isomorphism between the basic cohomology rings Hbas*(\documentclass[12pt]{minimal}
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W_\mathfrak{g}
$$\end{document}) and Hbas*([graphic not available: see fulltext]). In this paper, we will interpret the quantization map [graphic not available: see fulltext] as the super Duflo map between the symmetric algebra \documentclass[12pt]{minimal}
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S\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right)
$$\end{document} and the universal enveloping algebra \documentclass[12pt]{minimal}
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U\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right)
$$\end{document} of a super Lie algebra \documentclass[12pt]{minimal}
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\widetilde{T\mathfrak{g}\left[ 1 \right]}
$$\end{document} which is canonically associated with the quadratic Lie algebra \documentclass[12pt]{minimal}
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\mathfrak{g}
$$\end{document}. The basic cohomology rings Hbas*(\documentclass[12pt]{minimal}
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W_\mathfrak{g}
$$\end{document}) and Hbas*([graphic not available: see fulltext]) correspond exactly to \documentclass[12pt]{minimal}
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S\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right)^{inv}
$$\end{document} and \documentclass[12pt]{minimal}
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U\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right)^{inv}
$$\end{document}, 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.
