Left-invariant conformal vector fields on non-solvable Lie groups

Abstract:

Let $(G,\langle \cdot ,\cdot \rangle )$ be a pseudo-Riemannian Lie group of type $(p,q)$ with the Lie algebra $\mathfrak {g}$. In this paper, we prove that $G$ is solvable if $(G,\langle \cdot ,\cdot \rangle )$ admits a non-Killing left-invariant conformal vector field and $\dim [\mathfrak {g},\mathfrak {g}]=\dim \mathfrak {g}-\min (p,q)+1$ for $\min (p,q)\geq 2$. Then we construct a non-solvable pseudo-Riemannian Lie group $G$ of type $(p,q)$ which admits a non-Killing left-invariant conformal vector field and $\dim [\mathfrak {g},\mathfrak {g}]=\dim \mathfrak {g}-\min (p,q)+d$ for any $p,q\geq 3$ and $2\leq d\leq \min (p,q)-1$. It gives a negative answer to a forthcoming conjecture by H. Zhang and Z. Chen.