Superalgebraic interpretation of quantization maps of Weil algebras

Abstract

In 1998, A.Alekseev and E.Meinrenken construct an explicit $G$-differential space homomorphism $\mathcal{Q}$, called the quantization map, between the Weil algebra $\Weil{\g}= \sym{\co{\g}} \otimes \ext{\co{\g}}$ and $\NWeil{\g}=\U{\g} \otimes \Cl{\g}$ (which they called the noncommutative Weil algebra) for any quadratic Lie algebra $\g$. They showed that $\mathcal{Q}$ induces an algebra isomorphism between the basic cohomology rings $H^{\ast}_{bas}(\Weil{g})$ and $H^{\ast}_{bas}(\NWeil{\g})$. In this paper, I interpret the quantization map $\mathcal{Q}$ as the super Duflo map between the symmetric algebra $S(\widetilde{T\g[1]})$ and the universal enveloping algebra $U(\widetilde{T\g[1]})$ of a super Lie algebra $\widetilde{T\g[1]}$ which is canonically related to the quadratic Lie algebra $\g$. The basic cohomology rings $H^{\ast}_{bas}(\Weil{g})$ and $H^{\ast}_{bas}(\NWeil{g})$ correspond exactly to $S(\widetilde{T\g[1]})^{inv}$ and $U(\widetilde{T\g[1]})$ respectively. So what they proved is equivalent to the fact that the Duflo map commutes with the adjoint action of the Lie algebra, and that the Duflo map is an algebra homomorphism when restricted to the space of invariants. In addition, I will explain how the diagrammatic analogue of the Duflo map can be also made for the quantization map $\mathcal{Q}$.