Tensor products of quantized tilting modules

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.