This talk is concerned with the existence of a Dixmier map for nilpotent super Lie algebras
and its applications to the representation theory of super Yang-Mills algebras.
More precisely, we shall state results concerning the Kirillov orbit method a la Dixmier
for nilpotent super Lie algebras, i.e. that the usual Dixmier map for nilpotent Lie algebras
can be naturally extended to the context of nilpotent super Lie algebras. Moreover, our
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 (solvable) super Lie
algebras, generalizing the concept in the nonsuper case.
As a corollary of the previous description, we obtain that the quotient of the enveloping
algebra of a nilpotent super Lie algebra by a maximal ideal is isomorphic to the tensor product
of a Clifford algebra and a Weyl algebra, and we determine explicitely the indices of both of
them, we get a direct construction of the maximal ideals of the underlying
algebra of enveloping algebra and also some properties of the stabilizers of the primitive ideals.
All of these results can be used to study the representation theory of super algebras related to
the super Yang-Mills theory of interest in physics.