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Integral modular categories and integrality of quantum invariants at roots of unity of prime order. (English) Zbl 0919.57010

It is shown how to deduce integrality properties of quantum 3-manifold invariants from the existence of integral subcategories of modular categories. The method is illustrated in the case of the invariants associated to classical Lie algebras constructed by V. Turaev and H. Wenzl [Int. J. Math. 4, No. 2, 323-358 (1993; Zbl 0784.57007)], showing that the invariants are algebraic integers provided the root of unity has prime order. This generalizes results of H. Murakami [Math. Proc. Camb. Philos. Soc. 115, No. 2, 253-281 (1994; Zbl 0832.57005); 117, No. 2, 237-249 (1995; Zbl 0854.57016)] and of G. Masbaum and J. D. Roberts [ibid. 121, No. 3, 443-454 (1997; Zbl 0882.57010)] in the \(sl_2\)-case. One also discusses some details in the construction of invariants of 3-manifolds, such as the \(S\)-matrix in the \(PSU_k\) case, and a local orientation reversal principle for the colored Homfly polynomial.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
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[1] Murakami, Camb Phil The Kauffman polynomial of links and representation theory Osaka J Ram and Wenzl Matrix units for centralizer algebras Alg Reshetikhin Turaev Invariants of - manifolds via link polynomials and quantum groups Invent, Soc Math pp 117– (1995)
[2] Kauffman, Masbaum and Wenzl Quantum invariants An invariant of regular isotopy Trans AMS Wenzl Reconstructing monoidal categories Adv Soviet Math Part A calculus for framed links Invent The - manifold invariants of Witten and Reshetikhin - Turaev for si Invent, Math 2 pp 235– (1990)
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