On the finite-dimensional representations of the double of the Jordan plane

We continue the study of the Drinfeld double of the Jordan plane, denoted by $\mathcal{D}$ and introduced in arXiv:2002.02514. The simple finite-dimensional modules were computed in arXiv:2108.13849; it turns out that they factorize through $U({\text{spl}}_2(\mathbb{k}))$. Here we introduce the Verma modules and the category $\mathfrak{O}$ for $\mathcal{D}$, which have a resemblance to the similar ones in Lie theory but induced from indecomposable modules of the 0-part of the triangular decomposition. Accordingly, there is the notion of highest weight rank (hw-rk). We classify the indecomposable modules of hw-rk one and find families of hw-rk two. The Gabriel quiver of $\mathcal{D}$ is computed implying that it has a wild representation type.