  The partition functions of a 2D conformal system - the modular invariant one
or the generalized ones, coming from the introduction of defect lines - are
expressed in terms of a set of coefficients that have the particularity to form
nimreps of certain algebras. These coefficients define the various structure
maps of a new class of Hopf Algebras, called Weak Hopf Algebras, and can be
encoded in a set of graphs. The aim of chapter 1 is a presentation of the
actual knowledge on this subject. In chapter 2 the Weak Hopf Algebra together
with its structures is introduced, in particular the Ocneanu algebra of quantum
symmetries, that play a key role in the study of 2d conformal systems. We
analyze in details these structures for the A3 diagram, associated to the
affine su(2) conformal system. In chapter 3, we present a realization of the
Ocneanu algebra of quantum symmetries, constructed as a quotient of the tensor
square of a graph algebra (ADE graph for the affine (2) model). This
realization allows obtaining a very simple algorithm for the determination of
the partition functions associated with the conformal model. Our construction
can be naturally generalized to the affine su(n) cases, with n > 2, where few
results were known. All the cases related of su(2)-type as well as three
particular examples of su(3)-type are explicitly treated in chapter 4.
