Representation theory of super Yang-Mills algebras

We study in this article the representation theory of a family of super algebras, called the \emph{super Yang-Mills algebras}, by exploiting the Kirillov orbit method $\grave{\mathit{\text{a}}}\mathit{\text{la}}\mathit{\text{Dixmier}}$ for nilpotent super Lie algebras. These super algebras are a generalization of the so-called \emph{Yang-Mills algebras}, introduced by A. Connes and M. Dubois-Violette in \cite{CD02}, but in fact they appear as a “background independent” formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras ${\text{Cliff}}_q(k)\otimes A_p(k)$, for $p\ge 3$, or $p=2$ and $q\ge 2$, appear as a quotient of all super Yang-Mills algebras, for $n\ge 3$ and $s\ge 1$. This provides thus a family of representations of the super Yang-Mills algebras.