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In this work we discuss the elements required for the construction of theoperator algebra for the space of paths over a simply laced $SU(3)$ graph.These operators are an important step in the construction of the bialgebrarequired to find the partition functions of some modular invariant CFTs. Wedefine the cup and cap operators associated with back-and-forth sequences andadd them to the creation and annihilation operators in the operator algebra asthey are required for the calculation of the full space of essential pathsprescribed by the fusion algebra. These operators require collapsed triangularcells that had not been found in previous works; here we provide explicitvalues for these cells and show their importance in order for the cell systemto fulfill the Kuperberg relations for $SU(3)$ tangles. We also find thatdemanding that our operators satisfy the Temperley-Lieb algebra leads onenaturally to consider operators that create and annihilate closed triangularsequences, which in turn provides an alternative the cup and cap operators asthey allow one to replace back-and-forth sequences with closed triangular ones.We finally show that the essential paths obtained by using closed triangles areequivalent to those obtained originally using back-and-forth sequences.

Alternative formulation for the operator algebra over the space of paths in a ADE $SU(3)$ graph