  In this paper, we investigated the behavior of left-invariant conformal
vector fields on Lie groups with left-invariant pseudo-Riemannian metrics.
First of all, we prove that conformal vector fields on pseudo-Riemannian
unimodular Lie groups are Killing. Then we obtain a necessary condition for a
pseudo-Rimennian non-unimodular Lie group admitting a non-Killing conformal
vector field. Finally, we give examples of non-Killing conformal vector fields
and Yamabe solitons on non-unimodular Lorentzian Lie groups based on the above
study.
