\(C^{*}\) tensor categories from quantum groups. (English) Zbl 0888.46043
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 \(\text{II}_1\) factor for each object of \(\mathcal F\).
MSC:
| 46L37 | Subfactors and their classification |
| 46L60 | Applications of selfadjoint operator algebras to physics |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
Keywords:
quantum groups; modular tensor categories; quotient category; category of tilting modules; subfactor of the hyperfinite \(\text{II}_ 1\) factorReferences:
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