The classification of Lorentzian Lie groups with non-Killing left-invariant conformal vector fields

This paper is to classify Lorentzian Lie groups (G,⟨center dot,center dot⟩)$(G,\langle \cdot ,\cdot \rangle )$ admitting non-Killing left-invariant conformal vector fields whose Lie algebra is g${\mathfrak {g}}$. As we know, g${\mathfrak {g}}$ is solvable and [g,g]$[{\mathfrak {g}},{\mathfrak {g}}]$ is of codimension 1 in g${\mathfrak {g}}$. In this paper, we will prove that [g,g]$[{\mathfrak {g}},{\mathfrak {g}}]$ is an Abelian Lie algebra, or a direct sum of a generalized Heisenberg Lie algebra and an Abelian Lie algebra. We obtain a simple criterion for such Lorentzian Lie groups with dimG > 4$\dim G\geqslant 4$ to be conformally flat, and moreover, examples of conformally flat and non-conformally flat Lorentzian Lie groups are constructed. Finally, we prove that Lorentzian Lie groups admitting non-Killing left-invariant conformal vector fields can be shrinking, steady and expanding Ricci solitons.