Ground state hyperfine structure in muonic lithium ions

On the basis of perturbation theory in fine structure constant alpha and the ratio of electron to muon masses we calculate one-loop vacuum polarization, electron vertex corrections, nuclear structure and recoil corrections to hyperfine splitting of the ground state in muonic lithium ions $(\mu\ e\ ^6_3Li)^+$ and $(\mu\ e\ ^7_3Li)^+$. We obtain total results for the ground state small hyperfine splittings in $(\mu\ e\ ^6_3Li)^+$ $\Delta\nu_1=21572.16$ MHz and $\Delta\nu_2=14152.56$ MHz and in $(\mu\ e\ ^7_3Li)^+$ $\Delta\nu_1=21733.06$ MHz and $\Delta\nu_2=13994.35$ MHz which can be considered as a reliable estimate for a comparison with future experimental data.