  We introduce a systematic construction of a gapless symmetry protected
topological phase in one dimension by "decorating" the domain walls of
Luttinger liquids. The resulting strongly interacting phases provide a concrete
example of a gapless symmetry protected topological (gSPT) phase with robust
symmetry-protected edge modes. Using boundary conformal field theory arguments,
we show that while the bulks of such gSPT phases are identical to conventional
Luttinger liquids, their boundary critical behavior is controlled by a
different, strongly-coupled renormalization group fixed point. Our results are
checked against extensive density matrix renormalization group calculations.
