The Dixmier map for nilpotent super Lie algebras

Estanislao Herscovich

Published 2010-09-06, updated 2011-04-19Version 3

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).