The Dixmier Map for Nilpotent Super Lie Algebras

In this article we prove that there exists a Dixmier map for nilpotent super Lie algebras. In other words, if we denote by Prim(U(g)) the set of (graded) primitive ideals of the enveloping algebra U(g) of a nilpotent Lie superalgebra g and Ad0 the adjoint group of g0 , we prove that the usual Dixmier map for nilpotent Lie algebras can be naturally extended to the context of nilpotent super Lie algebras, i.e. there exists a bijective map I : g∗ 0/Ad0 → Prim(U(g)) defined by sending the equivalence class [λ] of a functional λ to a primitive ideal I (λ) of U(g), and which coincides with the Dixmier map in the case of nilpotent Lie algebras. Moreover, the construction of the previous map is explicit, and more or less parallel to the one for Lie algebras, a major difference with a previous approach (cf. [18]). One key fact in the construction is the existence of polarizations for super Lie alge- bras, generalizing the concept defined for Lie algebras. As a corollary of the previ- ous description, we obtain the isomorphism U(g)/I (λ) Cliffq (k) ⊗ A p(k), where ( p, q) = (dim(g0/gλ 0 )/2, dim(g1/gλ 1 )), we get a direct construction of the maximal ide- als of the underlying algebra of U(g) and also some properties of the stabilizers of the primitive ideals of U(g).