The Dixmier map for nilpotent super Lie algebras

TL;DR

This paper extends the classical Dixmier map to nilpotent super Lie algebras, providing an explicit bijection between certain dual space quotients and primitive ideals of the enveloping algebra, with applications to algebra structure and stabilizers.

Contribution

It introduces a natural, explicit Dixmier map for nilpotent super Lie algebras, generalizing the classical case and utilizing polarizations for super Lie algebra structure.

Findings

Existence of a bijective Dixmier map for nilpotent super Lie algebras.

Construction of primitive ideals as tensor products of Clifford and Weyl algebras.

Characterization of stabilizers and maximal ideals in the enveloping algebra.

Abstract

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 g and Ad_0 the adjoint group of 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 \[ I : g_0^*/Ad_0 \rightarrow Prim(U(g)), \] defined by sending the equivalence class [lambda] of a functional lambda to a primitive ideal I(lambda) 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. 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 that the quotient of the enveloping algebra by the ideal I(lambda) is isomorphic to the tensor product Cliff_q(k) \otimes A_p(k) of a Clifford algebra and a Weyl algebra, where (p,q) = (dim(g_0/g_0^{lambda})/2,dim(g_1/g_1^{lambda})), we get a direct construction of the maximal ideals of the underlying algebra of U(g) and also some properties of the stabilizers of the primitive ideals of U(g).