Diagrammatic Construction of Representations of Small Quantum $\mathfrak{sl}_2$

TL;DR

This paper introduces a diagrammatic combinatorial framework for representing the monoidal category generated by the fundamental representation of the small quantum group of rak{sl}_2 at roots of unity, extending the Temperley-Lieb category.

Contribution

It provides a novel diagrammatic construction of small quantum group representations, enhancing understanding of their categorical structure at roots of unity.

Findings

Developed a combinatorial description of the monoidal category

Extended the Temperley-Lieb category at elta = -q - q^{-1}

Facilitated diagrammatic analysis of small quantum group representations

Abstract

We provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of $\mathfrak{sl}_2$ at a root of unity $q$ of odd order. Our approach is diagrammatic, and it relies on an extension of the Temperley-Lieb category specialized at $\delta = -q-q^{-1}$.