The Dixmier Map for Nilpotent Super Lie Algebras

Abstract

In this article we prove that there exists a Dixmier map for nilpotent super Lie algebras. In other words, if we denote by ${\mathrm{Prim}(\mathcal{U}(\mathfrak{g}))}$ the set of (graded) primitive ideals of the enveloping algebra ${\mathcal{U}(\mathfrak{g})}$ of a nilpotent Lie superalgebra ${\mathfrak{g}}$ and ${\mathcal{A}d_{0}}$ the adjoint group of ${\mathfrak{g}_{0}}$, 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

$$I : \mathfrak{g}_{0}^{*}/\mathcal{A}d_{0} \rightarrow \mathrm{Prim}(\mathcal{U}(\mathfrak{g}))$$

defined by sending the equivalence class [λ] of a functional λ to a primitive ideal I(λ) of ${\mathcal{U}(\mathfrak{g})}$, 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 algebras, generalizing the concept defined for Lie algebras. As a corollary of the previous description, we obtain the isomorphism ${\mathcal{U}(\mathfrak{g})/I(\lambda) \simeq {\rm Cliff}_{q}(k) \otimes A_{p}(k)}$, where ${(p,q) = ({\rm dim}(\mathfrak{g}_{0}/\mathfrak{g}_{0}^{\lambda})/2,{\rm dim}(\mathfrak{g}_{1}/\mathfrak{g}_{1}^{\lambda}))}$, we get a direct construction of the maximal ideals of the underlying algebra of ${\mathcal{U}(\mathfrak{g})}$ and also some properties of the stabilizers of the primitive ideals of ${\mathcal{U}(\mathfrak{g})}$.