{"title":"On Lie Groups with Conformal Vector Fields Induced by Derivations","authors":"","doi":"10.1007/s00031-024-09845-4","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>A pseudo-Riemannian Lie group <span> <span>\\((G,\\langle \\cdot ,\\cdot \\rangle )\\)</span> </span> is a connected and simply connected Lie group with a left-invariant pseudo-Riemannian metric of signature (<em>p</em>, <em>q</em>). This paper is to study pseudo-Riemannian Lie group <span> <span>\\((G,\\langle \\cdot ,\\cdot \\rangle )\\)</span> </span> with conformal vector fields induced by derivations. Firstly, we show that if <span> <span>\\(\\mathfrak {h}\\)</span> </span> is a Cartan subalgebra for a semisimple Levi factor of <span> <span>\\({\\mathfrak g}\\)</span> </span>, where <span> <span>\\({\\mathfrak g}\\)</span> </span> denotes the Lie algebra of <em>G</em>, then <span> <span>\\(\\dim \\mathfrak {h}\\le \\max \\{0,\\min \\{p,q\\}-1\\}\\)</span> </span>. It implies that <span> <span>\\({\\mathfrak g}\\)</span> </span> is solvable for both Riemannian (i.e., <span> <span>\\(\\min \\{p,q\\}=0\\)</span> </span>) and Lorentzian (i.e., <span> <span>\\(\\min \\{p,q\\}=1\\)</span> </span>) cases, and furthermore we prove that <span> <span>\\(\\mathfrak {sl}_2(\\mathbb {R})\\)</span> </span> is the only possible Levi factor for the trans-Lorentzian (i.e., <span> <span>\\(\\min \\{p,q\\}=2\\)</span> </span>) case. Secondly, based on the classification of the Riemannian and Lorentzian cases in (Corrigendum J. Algebra <strong>603</strong>, 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. <strong>33</strong>, 1087–1093 1992), are isometric to Lorentzian Lie groups with conformal vector fields induced by derivations.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"308 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transformation Groups","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00031-024-09845-4","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍:
Transformation Groups will only accept research articles containing new results, complete Proofs, and an abstract. Topics include: Lie groups and Lie algebras; Lie transformation groups and holomorphic transformation groups; Algebraic groups; Invariant theory; Geometry and topology of homogeneous spaces; Discrete subgroups of Lie groups; Quantum groups and enveloping algebras; Group aspects of conformal field theory; Kac-Moody groups and algebras; Lie supergroups and superalgebras.