$C^{*}$ tensor categories from quantum groups

Let $\mathfrak g$ be a semisimple Lie algebra and let $d$ be the ratio between the square of the lengths of a long and a short root. Moreover, let $\mathcal F$ be the quotient category of the category of tilting modules of $U_q\mathfrak g$ modulo the ideal of tilting modules with zero $q$-dimension for $q=e^{\pm\pi i/dl}$. We show that for $l$ a sufficiently large integer, the morphisms of $\mathcal F$ are Hilbert spaces satisfying functorial properties. As an application, we obtain a subfactor of the hyperfinite II$_1$ factor for each object of $\mathcal F$.