Superalgebraic Interpretation of the Quantization Maps of Weil Algebras

In [2], A.Alekseev and E.Meinrenken construct an explicit G-differential space homomorphism Q, called the quantization map, between the Weil algebra Wg = S(g ) ⊗ ∧(g) and Wg = U (g) ⊗ Cl(g) (which they called the noncommutative Weil algebra) for any quadratic Lie algebra g. They showed that Q induces an algebra isomorphism between the basic cohomology rings H bas(Wg) and H ∗ bas(Wg). In this paper, I will interpret the quantization map Q as the super Duflo map between the symmetric algebra S(T̃g[1]) and the universal enveloping algebra U(T̃g[1]) of a super Lie algebra T̃g[1] which is canonically related to the quadratic Lie algebra g. The basic cohomology rings H bas(Wg) and H bas(Wg) correspond exactly to S(T̃g[1]) inv and U(T̃g[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 introduced in [6] can be also made for the quantization map Q. Acknowledgements: First of all, I would like to thank my advisor J.Roberts and professor N. Wallach for their support and insightful advices. I would also like to thank professor E. Meinrenken for pointing out some mistakes I wrote in the draft. In addition, the argument in section 5.5 is due to Dylan Thurston. I also understand that several people including E. Meinrenken and P. Severa have already conjectured the main results of this paper and probably have their way of proving them. However, no formal proof had yet been published. So I think it is worth writing it out. Date: May 1, 2005. Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112, USA. Email: [email protected].