For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the corresponding twisted partition functions.We dedicate this article to the memory of our friend Prof. Juan A. Mignaco deceased, 6 June 2001.
This is a set of notes describing several aspects of the space of paths on ADE Dynkin diagrams, with a particular attention paid to the graph E 6 . Many results originally due to A. Ocneanu are here described in a very elementary way (manipulation of square or rectangular matrices). We recall the concept of essential matrices (intertwiners) for a graph and describe their module properties with respect to right and left actions of fusion algebras. In the case of the graph E 6 , essential matrices build up a right module with respect to its own fusion algebra but a left module with respect to the fusion algebra of A 11 . We present two original results: 1) Our first contribution is to show how to recover the Ocneanu graph of quantum symmetries of the Dynkin diagram E 6 from the natural multiplication defined in the tensor square of its fusion algebra (the tensor product should be taken over a particular subalgebra); this is the Cayley graph for the two generators of the twelve dimensional algebra E 6 ⊗ A3 E 6 (here A 3 and E 6 refer to the commutative fusion algebras of the corresponding graphs). 2) To every point of the graph of quantum symmetries one can associate a particular matrix describing the " torus structure" of the chosen Dynkin diagram; following Ocneanu, one obtains in this way, in the case of E 6 , twelve such matrices of dimension 11 × 11, one of them is a modular invariant and encodes the partition function of the corresponding conformal field theory. Our own next contribution is to provide a simple algorithm for the determination of these matrices.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.