The braiding for representations of q-deformed affine $sl_2$

We compute the braiding for the `principal gradation of $U_q(hat{{it sl}_2})$ for $|q|=1$ from first principles, starting from the idea of a rigid braided tensor category. It is not necessary to assume either the crossing or the unitarity condition from S-matrix theory. We demonstrate the uniqueness of the normalisation of the braiding under certain analyticity assumptions, and show that its convergence is critically dependent on the number-theoretic properties of the number $tau$ in the deformation parameter $q=e^{2pi itau}$. We also examine the convergence using probability, assuming a uniform distribution for $q$ on the unit circle.