𝐶* tensor categories from quantum groups

Abstract

Let g \mathfrak g be a semisimple Lie algebra and let d d be the ratio between the square of the lengths of a long and a short root. Moreover, let F \mathcal F be the quotient category of the category of tilting modules of U q g U_q\mathfrak g modulo the ideal of tilting modules with zero q q -dimension for q = e ± π i / d l q=e^{\pm \pi i/dl} . We show that for l l a sufficiently large integer, the morphisms of F \mathcal F are Hilbert spaces satisfying functorial properties. As an application, we obtain a subfactor of the hyperfinite II 1 _1 factor for each object of F \mathcal F .