  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.
