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

Abstract

This paper is to classify Lorentzian Lie groups ( G , ⟨ · , · ⟩ ) $(G,\langle \cdot ,\cdot \rangle )$ 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 dim G ⩾ 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.