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

TL;DR

This paper explores the representation theory of the Drinfeld double of the Jordan plane, introducing Verma modules and classifying indecomposables, revealing the algebra's wild representation type.

Contribution

It introduces Verma modules and the category al for the double of the Jordan plane, and classifies indecomposable modules of certain highest weight ranks.

Findings

Simple modules factor through U(sl_2)

Classification of indecomposables of hw-rk one

Gabriel quiver shows wild representation type

Abstract

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 $U(\spl_2(\Bbbk))$. 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.