The braiding for representations of q-deformed affine sl2sl_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 $q=e^{2\pi i\tau}$. We also examine the convergence using probability, assuming a uniform distribution for $q$ on the unit circle.Comment: LaTeX, 10 pages with 2 figs, uses epsfi

Making nontrivially associated modular categories from finite groups

We show that the non-trivially associated tensor category constructed from left coset representatives of a subgroup of a finite group is a modular category. Also we give a definition of the character of an object in a ribbon category which is the category of representations of a braided Hopf algebra in the category. The definition is shown to be adjoint invariant and multiplicative. A detailed example is given. Finally we show an equivalence of categories between the non-trivially associated double D and the category of representations of the double of the group D(X).Comment: Approx 43 pages, uses LaTeX picture environmen