TENSOR-PRODUCTS OF QUANTIZED TILTING MODULES

Let U(k) denote the quantized enveloping algebra corresponding to a finite dimensional simple complex Lie algebra L. Assume that the quantum parameter is a root of unity in k of order at least the Coxeter number for L. Also assume that this order is odd and not divisible by 3 if type G2 occurs. We demonstrate how one can define a reduced tensor product on the family F consisting of those finite dimensional simple U(k)-modules which are deformations of simple L-modules 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.