  A quantum spin Hall insulator is a two-dimensional state of matter consisting
of an insulating bulk and one-dimensional helical edge states. While these edge
states are topologically protected against elastic backscattering in the
presence of disorder, interaction-induced inelastic terms may yield a finite
conductivity. By using a kinetic equation approach, we find the backscattering
rate $\tau^{-1}$ and the semiclassical conductivity in the regimes of high
($\omega \gg \tau^{-1}$) and low ($\omega \ll \tau^{-1}$) frequency. By
comparing the two limits, we find that the parametric dependence of
conductivity is described by the Drude formula for the case of a disordered
edge. On the other hand, in the clean case where the resistance originates from
umklapp interactions, the conductivity takes a non-Drude form with a parametric
suppression of scattering in the dc limit as compared to the ac case. This
behavior is due to the peculiarity of umklapp scattering processes involving
necessarily the state at the "Dirac point". In order to take into account
Luttinger liquid effects, we complement the kinetic equation analysis by
treating interactions exactly in bosonization and calculating conductivity
using the Kubo formula. In this way, we obtain the frequency and temperature
dependence of conductivity over a wide range of parameters. We find the
temperature and frequency dependence of the transport scattering time in a
disordered system as $\tau \sim [\max{(\omega,T)}]^{-2K-2}$, for $K>2/3$ and
$\tau \sim [\max{(\omega,T)}]^{-8K+2}$ for $K <2/3$.
