  We demonstrate that the matrix quantum group $SL_q(2)$ gives rise to
nontrivial matrix product operator representations of the Lie group $SL(2)$,
providing an explicit characterization of the nontrivial global $SU(2)$
symmetry of the XXZ model with periodic boundary conditions. The matrix product
operators are non-injective and their set is closed under multiplication. This
allows to calculate the fusion tensors acting on the virtual or quantum degrees
of freedom and to obtain the recoupling coefficients, which satisfy a type of
pentagon relation. We argue that the combination of this data with the well
known $q$-deformed Clebsch-Gordan coefficients and 6j-symbols is consistent
with a description of this quantum group in terms of bimodule categories.
