Quantum group interpretation of some conformal field theories

We show that the representation theory of the q-deformation of SU(2) provides solutions to the polynomial equations of Moore and Seiberg for rational conformal field theories whenever q is a root of unity. The q-analogue of the 6j-symbols give the duality matrices, and there is a close connection between the modular properties of the Kaĉ-Moody character for SU(2)_k and some simple properties of the q-characters of quantum groups. We show how the quantum group can be considered for most purposes as a rather accurate description of the Wess-Zumino-Witten theory at level k, where k is determined by q.