Notes on the quantum tetrahedron

Abstract

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 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 E6, 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 A11. 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 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 and E6 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 E6, twelve such matrices of dimension 11x11, 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.