Abstract
LetU k denote the quantized enveloping algebra corresponding to a finite dimensional simple complex Lie algebraL. Assume that the quantum parameter is a root of unity ink of order at least the Coxeter number forL. Also assume that this order is odd and not divisible by 3 if typeG 2 occurs. We demonstrate how one can define a reduced tensor product on the familyF consisting of those finite dimensional simpleU k-modules which are deformations of simpleL and which have non-zero quantum dimension. This together with the work of Reshetikhin-Turaev and Turaev-Wenzl prove that (U k,F) is a modular Hopf algebra and hence produces invariants of 3-manifolds. Also by recent work of Duurhus, Jakobsen and Nest it leads to a general topological quantum field theory. The method of proof explores quantized analogues of tilting modules for algebraic groups.
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[APW1] Andersen, H.H., Polo, P., Wen, K.: Representations of quantum algebras. Invent. Math.104, 1–59 (1991)
[APW2] Andersen, H.H., Polo, P., Wen, K.: Injective modules for quantum algebras. Am. J. Math.114, 571–604 (1992)
[AW] Andersen, H.H., Wen, K.: Representations of quantum algebras. The mixed case. J. Reine Angew. Math.427, 35–50 (1992)
[CPSvdK] Cline, E., Parshall, B., Scott, L., van der Kallen, W.: Rational and generic cohomology. Invent. Math.39, 129–165 (1980)
[D1] Donkin, S.: Rational representations of algebraic groups. Lecture Notes in Math. vol.1140, Berlin Heidelberg, New York: Springer 1985
[D2] Donkin, S.: On tilting modules for algebraic groups. Preprint QMW (1991)
[DJN] Durhuus, B., Jakobsen, P., Nest, R.: Topological Quantum Field Theories from Generalized 6J-Symbols. Preprint No. 11 (1991), Københavns University
[J] Jantzen, J.C.: Representations of algebraic groups. Pure Appl. Math., vol.131, London, New York: Academic Press 1987
[KL] Kazhdan, D., Lusztig, G.: Affine Lie algebras and quantum groups. Internat. Math. Research Notes (Duke Math. J.)2, 21–29 (1991)
[KM] Kirby, R., Melvin, P.: The 3-manifold invariants of Witten and Reshetikhin-Turaev forsl (2,C). Invent. Math.105, 473–545 (1991)
[L1] Lusztig, G.: Quantum groups at roots of 1. Geom. Dedicata35, 89–114 (1990)
[L2] Lusztig, G.: Canonical bases arising from quantized enveloping algebras II. Progr. Phys. Suppl.102, 175–202 (1990)
[M] Mathieu, O.: Filtrations ofG-modules. Ann. Sci. École Norm. Sup.23, 625–644 (1990)
[P] Paradowski, J.: Filtrations of modules over quantum algebras. Preprint 1992
[R] Ringel, C.M.: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z.208, 209–223 (1991)
[RT] Reshetikhin, N., Turaev, V.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math.103, 547–597 (1991)
[T] Thams, L.: Two classical results in the quantum mixed case. J. Reine Angew. Math. (to appear)
[TV] Turaev, V., Viro, O.: State sum invariants of 3-manifolds and quantum 6j-symbols. Preprint 1991
[TW] Turaev, V., Wenzl, H.: Quantum invariants of 3-manifolds associated with classical simple Lie algebras. Preprint 1991
[W] Wang, J-p.: Sheaf cohomology ofG/B and tensor products of Weyl modules. J. Algebra77, 162–185 (1982)
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Communicated by K. Gawedzki
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Andersen, H.H. Tensor products of quantized tilting modules. Commun.Math. Phys. 149, 149–159 (1992). https://doi.org/10.1007/BF02096627
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DOI: https://doi.org/10.1007/BF02096627