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 E6. Many results originally due to A. Ocneanu are here described in a very elementary way (manipulation of square or rectangular matrices). We define the concept of essential matrices for a graph and describe their module properties with respect to right and left actions of fusion algebras. In the case of the graph E6, essential matrices build up a right module with respect to its fusion algebra but a left module with respect to the fusion algebra of A11. We present two original results: 1) We show how to recover the Ocneanu graph of quantum symmetries of the Dynkin diagram E6 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 (E6 \otimes_A3 E6); here A3 and E6 refer to the commutative fusion algebras of the corresponding graphs. 2) One already knows how to associate, with every point of the graph of quantum symmetries, a particular matrix describing the `` torus structure'' of the chosen Dynkin diagram (Ocneanu construction). In the case of E6, one obtains in this way twelve such matrices of dimension 11x11; one of them is a modular invariant and encodes the partition function of the corresponding conformal field theory. We introduce a very simple algorithm that allows one to compute these matrices.