  We analyze the properties of a Luttinger liquid under the influence of a
periodic driving of the interaction strength. Irrespective of the details the
driven system develops an instability due to a parametric resonance. For slow
and fast driving, however, we identify intermediate long-lived meta-stable
states at constant time-averaged internal energies. Due to the instability
perturbations in the fermionic density are amplified exponentially leading to
the buildup of a superlattice. The momentum distribution develops a terrace
structure due to scattering processes that can be associated with the
absorption of quanta of the driving frequency.
