For the SU (3) system of graphs generalizing the ADE Dynkin digrams in the classification of modular invariant partition functions in CFT, we present a general collection of algebraic objects and relations that describe fusion properties and quantum symmetries associated with the corresponding Ocneanu quantum groupoïds. We also summarize the properties of the individual members of this system.

The affine su(3) modular invariant partition functions in 2d RCFT are associated with a set of generalized Coxeter graphs. These partition functions fall into two classes, the block-diagonal (Type I) and the non block-diagonal (Type II) cases, associated, from spectral properties, to the subsets of subgroup and module graphs respectively. We introduce a modular operatorT taking values on the set of vertices of the subgroup graphs. It allows us to obtain easily the associated Type I partition functions. We also show that all Type II partition functions are obtained by the action of suitable twists ϑ on the set of vertices of the subgroup graphs. These twists have to preserve the values of the modular operatorT .