  We study spin transport between a ferromagnet with time-dependent
magnetization and a conducting carbon nanotube or quantum wire, modeled as a
Luttinger liquid. The precession of the magnetization vector of the ferromagnet
due for instance to an outside applied magnetic field causes spin pumping into
an adjacent conductor. Conversely, the spin injection causes increased
magnetization damping in the ferromagnet. We find that, if the conductor
adjacent to the ferromagnet is a Luttinger liquid, spin pumping/damping is
suppressed by interactions, and the suppression has clear Luttinger liquid
power law temperature dependence. We apply our result to a few particular
setups. First we study the effective Landau-Lifshitz-Gilbert (LLG) coupled
equations for the magnetization vectors of the two ferromagnets in a FM-LL-FM
junction. Also, we compute the Gilbert damping for a FM-LL and a FM-LL-metal
junction.
