A superalgebraic interpretation of the quantization maps of Weil algebras

Let G be a Lie group whose Lie algebra 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 and (which they call the noncommutative Weil algebra) for . They showed that induces an algebra isomorphism between the basic cohomology rings H _bas^*() and H _bas^*( ). In this paper, we will interpret the quantization map as the super Duflo map between the symmetric algebra and the universal enveloping algebra of a super Lie algebra which is canonically associated with the quadratic Lie algebra . The basic cohomology rings H _bas^*() and H _bas^*( ) correspond exactly to and , 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.