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An explicit and complete construction of the $SU(3)$ $A_1$ associated quantumgroupoid is presented in this work, inspired by the approach taken by Trincherofor the $SU(2)$ $A_l$ graphs. New creation and annihilation operators weredefined in order to consider the $3$ different types of back-tracks whichappear due to the specific structure of $SU(3)$. The $C^{\star}$ bialgebra andthe realization of a Temperley-Lieb algebra is studied thoroughly. Finally, itis 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 theresults of this work. The generalization for higher levels $A_l$ graphs arestill an unsolved challenge, but now we count with enough tools, some insightabout how to attack this problem, and the first steps towards solving it.

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