A pseudo-Riemannian Lie group (G,⟨center dot,center dot⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G,\langle \cdot ,\cdot \rangle )$$\end{document} is a connected and simply connected Lie group with a left-invariant pseudo-Riemannian metric of signature (p, q). This paper is to study pseudo-Riemannian Lie group (G,⟨center dot,center dot⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G,\langle \cdot ,\cdot \rangle )$$\end{document} with conformal vector fields induced by derivations. Firstly, we show that if h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {h}$$\end{document} is a Cartan subalgebra for a semisimple Levi factor of g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}$$\end{document}, where g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}$$\end{document} denotes the Lie algebra of G, then dimh <= max{0,min{p,q}-1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dim \mathfrak {h}\le \max \{0,\min \{p,q\}-1\}$$\end{document}. It implies that g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}$$\end{document} is solvable for both Riemannian (i.e., min{p,q}=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \{p,q\}=0$$\end{document}) and Lorentzian (i.e., min{p,q}=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \{p,q\}=1$$\end{document}) cases, and furthermore we prove that sl2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {sl}_2(\mathbb {R})$$\end{document} is the only possible Levi factor for the trans-Lorentzian (i.e. , min{p,q}=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \{p,q\}=2$$\end{document}) case. Secondly, based on the classification of the Riemannian and Lorentzian cases in (Corrigendum J. Algebra 603, 38-40 2022), we prove that the Riemannian Lie groups are of constant zero sectional curvature, hence conformally flat; for the Lorentzian case, we obtain a simple criterion for such Lorentzian Lie groups to be conformally flat, and moreover, we show that they are steady algebraic Ricci soliton with vanishing scalar curvature. Finally, we remark that the first known examples of homogeneous essential Lorentzian manifolds that are non-conformally flat (Translation in Siberian Math. J. 33, 1087-1093 1992), are isometric to Lorentzian Lie groups with conformal vector fields induced by derivations.
