Let \(U_ k\) be the quantized enveloping algebra of a finite-dimensional simple complex Lie algebra \(L\), \(k\) a field. Assume that the quantum parameter is a root of unity in \(k\) of order at least the Coxeter number of \(L\), that this order is odd, and not divisible by 3 if type \(G_ 2\) occurs. The author defines a reduced tensor product on the family \(F\) of those finite-dimensional simple \(U_ k\)-modules which are deformations of simple \(L\)-modules and have non-zero quantum dimension. Thus \((U_ k,F)\) is a modular Hopf algebra which produces invariants of 3-manifolds, and leads to a generalized quantum field theory. The author's methods involve quantized analogues of tilting modules for algebraic groups.