Prime spectrum and representations of the super Jordan plane

We study the ring-theoretic structure and representation theory of the super Jordan plane $\mathcal{J}$ over fields of characteristic different from $2$.

We prove that $\mathcal{J}$ is prime and classify its prime, primitive, and maximal ideals.

We determine its classical ring of quotients and classify the finite-dimensional simple modules, while relating infinite-dimensional simple modules to those of the first Weyl algebra.

Our approach is based on showing that a localization of $\mathcal{J}$ is a matrix algebra over a localization of the first Weyl algebra.