A diagrammatic definition of $U_q(sl_2)$

TL;DR

This paper provides a diagrammatic framework for defining the quantum group $U_q(sl_2)$, detailing its Hopf algebra structure and connections with the Temperley-Lieb category, for generic q.

Contribution

It introduces a novel diagrammatic approach to $U_q(sl_2)$, clarifying its structure and relationship with categorical models, extending previous algebraic definitions.

Findings

Diagrammatic definition of $U_q(sl_2)$ for non-root of unity q

Explicit Hopf algebra structure in diagrammatic form

Connection established with Temperley-Lieb category

Abstract

We give a diagrammatic definition of $U_q(sl_2)$ when $q$ is not a root of unity, including its Hopf algebra structure and its relationship with the Temperley-Lieb category.