  We study an interacting composite $(1+1/n)$ Abelian helical edge state made
of a regular helical liquid carrying charge $e$ and a (fractionalized) helical
liquid carrying charge $e/n$. A systematic framework is developed for these
composite $(1+1/n)$ Abelian helical edge states with $n=1,2,3$. For $n=2$, the
composite edge state consists of a regular helical Luttinger liquid and a
fractional topological insulator (the Abelian $Z_4$ topological order) edge
state arising from half-filled conjugated Chern bands. The composite edge state
with $n=2$ is pertinent to the recent twisted MoTe$_2$ experiment, suggesting a
possible fractional topological insulator with conductance
$\frac{3}{2}\frac{e^2}{h}$ per edge. Using bosonization, we construct generic
phase diagrams in the presence of $weak$ Rashba spin-orbit coupling. In
addition to a phase of free bosons, we find a time-reversal symmetry-breaking
localized insulator, two perfect positive drag phases, a perfect negative drag
phase (for $n=2,3$), a time-reversal symmetric Anderson localization (only for
$n=1$), and a disorder-dominated metallic phase analogous to the $\nu=2/3$
disordered fractional quantum Hall edges (only for $n=3$). We further compute
the two-terminal edge-state conductance, the primary experimental
characterization for the (fractional) topological insulator. Remarkably, the
negative drag phase gives rise to an unusual edge-state conductance,
$(1-1/n)\frac{e^2}{h}$, not directly associated with the filling factor. We
further investigate the effect of an applied in-plane magnetic field. For
$n>1$, the applied magnetic field can result in a phase with edge-state
conductance $\frac{1}{n}\frac{e^2}{h}$, providing another testable signature.
Our work establishes a systematic understanding of the composite $(1+1/n)$
Abelian helical edge, paving the way for future experimental and theoretical
studies.
