# Non-Semisimple Planar Algebras from the Representation Theory of $\bar{U}_{q}(\mathfrak{sl}_{2})$

## TL;DR

This paper constructs a planar algebra from the restricted quantum group U_q(sl_2) by describing generators and relations, providing a diagrammatic approach to endomorphism algebras of tensor powers of a 2D module.

## Contribution

It offers a new diagrammatic description of endomorphism algebras associated with U_q(sl_2), specifically for the non-semisimple case.

## Key findings

- Provides generators and relations for the planar algebra
- Describes the endomorphism algebra diagrammatically
- Connects quantum group representations with planar algebra structures

## Abstract

We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group $\bar{U}_{q}(\mathfrak{sl}_{2})$. This is a diagrammatic description of $End_{\bar{U}_{q}(\mathfrak{sl}_{2})}(X^{\otimes n})$, where $X:=\mathcal{X}^{+}_{2}$ is a two dimensional $\bar{U}_{q}(\mathfrak{sl}_{2})$ module.