  We study the transport properties for a Luttinger-liquid (LL) quantum wire in
the presence of both Rashba spin-orbit coupling (SOC) and a weak external
in-plane magnetic field. The bosonized Hamiltonian of the system with an
externally applied longitudinal electric field is established. And then the
equations of motion for the bosonic phase fields are solved in the Fourier
space, with which the both charge and spin conductivities for the system are
calculated analytically based on the linear response theory. Generally, the ac
conductivity is an oscillation function of the strengths of electron-electron
interaction, Rashba SOC and magnetic field, as well as the driving frequency
and the measurement position in the wire. Through analysis with some examples
it is demonstrated that the modification on the conductivity due to
electron-electron interactions is more remarkable than that due to SOC, while
the effects of SOC and Zeeman splitting on the conductivity are very similar.
The spin-polarized conductivities for the system in the absence of Zeeman
effect or SOC are also discussed, respectively. The ratio of the spin-polarized
conductivities $\sigma_\uparrow/\sigma_\downarrow$ is dependent of the
electron-electron interactions for the system without SOC, while it is
independent of the electron-electron interactions for the system without Zeeman
splitting.
