𝐶* tensor categories from quantum groups

Wenzl, Hans

doi:10.1090/S0894-0347-98-00253-7

$C^*$ tensor categories from quantum groups
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by Hans Wenzl

J. Amer. Math. Soc. 11 (1998), 261-282

DOI: https://doi.org/10.1090/S0894-0347-98-00253-7

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Abstract:

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 $\mathcal F$.

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