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


Author: Hans Wenzl
Journal: J. Amer. Math. Soc. 11 (1998), 261-282
MSC (1991): Primary 81R50, 46L37
DOI: https://doi.org/10.1090/S0894-0347-98-00253-7
MathSciNet review: 1470857
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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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Additional Information

Hans Wenzl
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112
Email: wenzl@brauer.ucsd.edu

DOI: https://doi.org/10.1090/S0894-0347-98-00253-7
Keywords: Quantum groups at roots of 1, modular tensor categories, subfactors
Received by editor(s): February 7, 1997
Received by editor(s) in revised form: September 29, 1997
Additional Notes: The author was supported in part by NSF grant # DMS 94-00987
Article copyright: © Copyright 1998 American Mathematical Society

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