  We define a natural quantum analogue for the ${\cal Z}$ algebra, and which we
refer to as the ${\cal Z}_q$ algebra, by modding out the Heisenberg algebra
from the quantum affine algebra $U_q(\hat{sl(2)})$ with level $k$. We discuss
the representation theory of this ${\cal Z}_q$ algebra. In particular, we
exhibit its reduction to a group algebra, and to a tensor product of a group
algebra with a quantum Clifford algebra when $k=1$, and $k=2$, and thus, we
recover the explicit constructions of $\uq$-standard modules as achieved by
Frenkel-Jing and Bernard, respectively. Moreover, for arbitrary nonzero level
$k$, we show that the explicit basis for the simplest ${\cal Z}$-generalized
Verma module as constructed by Lepowsky and primc is also a basis for its
corresponding ${\cal Z}_q$-module, i.e., it is invariant under the
q-deformation for generic q. We expect this ${\cal Z}_q$ algebra (associated
with $\uq$ at level $k$), to play the role of a dynamical symmetry in the
off-critical $ Z_k$ statistical models.
