Representations of tensor categories and Dynkin diagrams

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)}.