Yu LI
Let
G be a Lie group whose Lie algebra g is quadratic. In the paper "the non-commutative Weil algebra", Alekseev and Meinrenken constructed an explicit
G-differential space homomorphism
L , called the quantization map, between the Weil algebra W
g = S(g
*)⊕∧(g
*) and
Wg=
U(g)⊕
Cl(g) (which they call the noncommutative Weil algebra) for g . They showed that
L induces an algebra isomorphism between the basic cohomology rings
Hbas * (
Wg) and
Hbas * (
Wg). In this paper, we will interpret the quantization map
L as the super Duflo map between the symmetric algebra
S and the universal enveloping algebra
U of a super Lie algebra
which is canonically associated with the quadratic Lie algebra g . The basic cohomology rings
Hbas* (
Wg) and
Hbas* (
Wg) correspond exactly to
S and
U 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.