Abstract
Given any simple biorientable graph it is shown that there exists a weak
*-Hopf algebra constructed out of the space of paths on the corresponding
graph. This construction is based on a direct sum decomposition of the space of
paths into orthogonal subspaces one of which is the space of essential paths.
Two simple examples are worked out with certain detail, the ADE graph $A_3$
and the affine graph $A_2$. For the first example the weak *-Hopf algebra
coincides with the so called double triangle algebra. No use is made of
Ocneanu's cell calculus.
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