The Braiding for Representations of Q-deformed Affine Sl 2

We compute the braiding for the 'principal gradation' of U q (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 τ in the deformation parameter q = e 2πiτ. We also examine the convergence using probability, assuming a uniform distribution for q on the unit circle.