Prime ideals and representations of the bosonization of the super Jordan plane

We study the Hopf algebra $A$ given by the bosonization of the super Jordan plane over a field of characteristic not 2. We determine its centre, classical quotient ring, prime and primitive ideals. Over an algebraically closed base field, we classify all simple $A$-modules. In characteristic zero, simple modules are either finite-dimensional, of dimension 1 or 2, or infinite-dimensional, arising from a localization of the first Weyl algebra. Over an algebraically closed field of characteristic $p>2$, every simple module is finite-dimensional, of dimension 1, 2, or $2p$, and we give explicit representatives for all isomorphism classes. Our approach relies on realizing a suitable localization of $A$ as a matrix algebra.