  In this note we illustrate by a few examples the general principle:
interesting algebras and representations defined over Z_+ come from category
theory, and are best understood when their categorical origination has been
discovered. We show that indecomposable Z_+-representations of the character
ring of SU(2) satisfying certain conditions correspond to affine and infinite
Dynkin diagrams with loops. We also show that irreducible Z_+-representations
of the Verlinde algebra (the character ring of the quantum group SU(2)_q, where
q is a root of unity), satisfying similar conditions correspond to usual
(non-affine) Dynkin diagrams with loops. Conjecturedly, the last result is
related to the ADE classification of conformal field theories with the chiral
algebra \hat{sl(2)}.
