On Lie Groups with Conformal Vector Fields Induced by Derivations

Pub Date : 2024-02-06 DOI:10.1007/s00031-024-09845-4

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Abstract

A pseudo-Riemannian Lie group \((G,\langle \cdot ,\cdot \rangle )\) 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 \((G,\langle \cdot ,\cdot \rangle )\) with conformal vector fields induced by derivations. Firstly, we show that if \(\mathfrak {h}\) is a Cartan subalgebra for a semisimple Levi factor of \({\mathfrak g}\) , where \({\mathfrak g}\) denotes the Lie algebra of G, then \(\dim \mathfrak {h}\le \max \{0,\min \{p,q\}-1\}\) . It implies that \({\mathfrak g}\) is solvable for both Riemannian (i.e., \(\min \{p,q\}=0\) ) and Lorentzian (i.e., \(\min \{p,q\}=1\) ) cases, and furthermore we prove that \(\mathfrak {sl}_2(\mathbb {R})\) is the only possible Levi factor for the trans-Lorentzian (i.e., \(\min \{p,q\}=2\) ) 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.

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论衍生诱导的具有共形矢量场的李群

Abstract A pseudo-Riemannian Lie group （(G,\langle \cdot ,\cdot\rangle)）是一个具有左不变伪黎曼度量的签名为（p, q）的连通且简单连通的李群。本文将研究具有由导数诱导的共形向量场的伪黎曼李群（(G,\langle \cdot ,\cdot \rangle )）。首先，我们证明如果\(\mathfrak {h}\)是\({\mathfrak g}\)的半简单列维因子的笛卡尔子代数，其中\({\mathfrak g}\)表示G的李代数，那么\(\dim \mathfrak {h}\le\max \{0,\min \{p,q\}-1\}) .这意味着对于黎曼（即，（min）（p,q）=0）和洛伦兹（即、 \此外，我们还证明了 \(\mathfrak {sl}_2(\mathbb {R})\) 是反洛伦兹（即 \(\min\{p,q\}=2\) ）情况下唯一可能的 Levi 因子。其次，基于（Corrigendum J. Algebra 603, 38-40 2022）中对黎曼和洛伦兹情形的分类，我们证明了黎曼李群具有恒定的零截面曲率，因此是保角平坦的；对于洛伦兹情形，我们得到了此类洛伦兹李群是保角平坦的简单判据，此外，我们还证明了它们是具有消失标量曲率的稳定代数黎氏孤子。最后，我们指出，已知的第一个非共形平坦的同质本质洛伦兹流形的例子（译文见《西伯利亚数学杂志》33，1087-1093 1992），与具有由导数诱导的共形向量场的洛伦兹李群是等距的。

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