  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.
