# On the Bosonization of the Super Jordan Plane

## TL;DR

This paper studies the algebraic structures arising from bosonizations of the Jordan and super Jordan planes, classifies their simple modules, and explores their representation categories, revealing connections to quantum groups at roots of unity.

## Contribution

It introduces the bosonizations of the Jordan and super Jordan planes and classifies their simple modules, providing new insights into their representation theory.

## Key findings

- Classified all finite-dimensional simple modules over the algebras.
- Listed indecomposable modules of dimension up to 5.
- Described a monoidal subcategory of the representation category.

## Abstract

Let $H$ and $K$ be the bosonizations of the Jordan and super Jordan plane by the group algebra of a cyclic group; the algebra $K$ projects onto an algebra $L$ that can be thought of as the quantum Borel of $\mathfrak{sl}(2)$ at $-1$. The finite-dimensional simple modules over $H$ and $K$, are classified; they all have dimension $1$, respectively $\le 2$. The indecomposable $L$-modules of dimension $\leq 5$ are also listed. An interesting monoidal subcategory of $\operatorname{rep} L$ is described.