Notes on the quantum tetrahedron

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 described here 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 $E_6\otimes_{A_3}E_6$ (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 $11\times 11$, one of them is a modular invariant and encodes the partition function of which corresponding conformal field theory. Our own next contribution is to provide a simple algorithm for the determination of these matrices.