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 $(\mu e{}_3^{6}Li{)}^{+}$ $\mathrm{\Delta }{\nu }_1=21572.16$ MHz and $\mathrm{\Delta }{\nu }_2=14152.56$ MHz and in $(\mu e{}_3^{7}Li{)}^{+}$ $\mathrm{\Delta }{\nu }_1=21733.06$ MHz and $\mathrm{\Delta }{\nu }_2=13994.35$ MHz which can be considered as a reliable estimate for a comparison with future experimental data.