In this paper we describe quantizations in the monoidal categories of unitary representations of compact connected Lie groups. For the n-dimensional torus T we show that the set Q(T ) of quantizations is isomorphic to the  n 2  -dimensional torus. For connected compact Lie groups G of rank n, we get the result that the set QE (G) of extendible quantizations of G-modules is isomorphic to the set of quantizations of its maximal torus T invariant under action by its Weyl group. For all these cases we give explicit formulae for quantizations and apply these to quantize Hilbert–Schmidt operators.