  We study a Luttinger Liquid (LL) coupled to a generic environment consisting
of bosonic modes with arbitrary density-density and current-current
interactions. The LL can be either in the conducting phase and perturbed by a
weak scatterer or in the insulating phase and perturbed by a weak link. The
environment modes can also be scattered by the imperfection in the system with
arbitrary transmission and reflection amplitudes. We present a general method
of calculating correlation functions under the presence of the environment and
prove the duality of exponents describing the scaling of the weak scatterer and
of the weak link. This duality holds true for a broad class of models and is
sensitive to neither interaction nor environmental modes details, thus it shows
up as the universal property. It ensures that the environment cannot generate
new stable fixed points of the RG flow. Thus, the LL always flows toward either
conducting or insulating phase. Phases are separated by a sharp boundary which
is shifted by the influence of the environment. Our results are relevant, for
example, for low-energy transport in (i) an interacting quantum wire or a
carbon nanotube where the electrons are coupled to the acoustic phonons
scattered by the lattice defect; (ii) a mixture of interacting fermionic and
bosonic cold atoms where the bosonic modes are scattered due to an abrupt local
change of the interaction, (iii) mesoscopic electric circuits.
