# The $SU(3)$ $A_1$ graph and its associated quantum groupoid

## TL;DR

This paper constructs the quantum groupoid associated with the $SU(3)$ $A_1$ graph, extending methods from $SU(2)$, and provides tools for generalizing to higher $SU(N)$ graphs, advancing the understanding of quantum symmetries.

## Contribution

It offers an explicit construction of the $SU(3)$ $A_1$ quantum groupoid, introducing new operators and analyzing its algebraic structures, paving the way for generalizations to higher levels.

## Key findings

- Explicit construction of the $SU(3)$ $A_1$ quantum groupoid.
- Development of new creation and annihilation operators for $SU(3)$.
- Framework for extending to $SU(N)$ graphs and higher levels.

## Abstract

An explicit and complete construction of the $SU(3)$ $A_1$ associated quantum groupoid is presented in this work, inspired by the approach taken by Trinchero for the $SU(2)$ $A_l$ graphs. New creation and annihilation operators were defined in order to consider the $3$ different types of back-tracks which appear due to the specific structure of $SU(3)$. The $C^{\star}$ bialgebra and the realization of a Temperley-Lieb algebra is studied thoroughly. Finally, it is shown that the construction of the quantum groupoids associated to the $A_{1}$ $SU(N)$ graphs are easily obtained for any value of $N$ using the results of this work. The generalization for higher levels $A_l$ graphs are still an unsolved challenge, but now we count with enough tools, some insight about how to attack this problem, and the first steps towards solving it.