  For quantized universal enveloping algebras we construct weight modules by
inducing representations of the centralizer of the Cartan subalgebra in the
quantized universal enveloping algebra. The induced modules arising from
finite-dimensional weight modules the centralizer algebra are studied. In
particular, we study the induction of one-dimensional modules, and this is
related to the study of commutative subalgebras of the centralizer algebra. For
the special case of $U_q(\mathfrak{sl}(2,\mathbb{C}))$ we show that we get the
admissible unitary representations corresponding to the non-compact real form
$U_q(\mathfrak{su}(1,1))$.
