  Molybdenum purple bronze Li$_{0.9}$Mo$_{6}$O$_{17}$ is an exceptional
material known to exhibit one dimensional (1D) properties for energies down to
a few meV. This fact seems to be well established both in experiments and in
band structure theory. We use the unusual, very 1-dimensional band dispersion
obtained in \emph{ab-initio} DFT-LMTO band calculations as our starting point
to study the physics emerging below 300meV. A dispersion perpendicular to the
main dispersive direction is obtained and investigated in detail. Based on
this, we derive an effective low energy theory within the Tomonaga Luttinger
liquid (TLL) framework. We estimate the strength of the possible interactions
and from this deduce the values of the TLL parameters for charge modes. Finally
we investigate possible instabilities of TLL by deriving renormalization group
(RG) equations which allow us to predict the size of potential gaps in the
spectrum. While $2k_F$ instabilities strongly suppress each other, the $4k_F$
instabilities cooperate, which paves the way for a possible CDW at the lowest
energies. The aim of this work is to understand the experimental findings, in
particular the ones which are certainly lying within the 1D regime. We discuss
the validity of our 1D approach and further perspectives for the lower energy
phases.
