1. Conformal vector fields on Lie groups: The trans-Lorentzian signature.
- Author
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Zhang, Hui, Chen, Zhiqi, and Tan, Ju
- Subjects
- *
LIE groups , *SEMISIMPLE Lie groups , *LIE algebras , *FACTORS (Algebra) , *VECTOR fields , *CURVATURE - Abstract
A pseudo-Riemannian Lie group is a connected Lie group endowed with a left-invariant pseudo-Riemannian metric of signature (p , q). In this paper, we study pseudo-Riemannian Lie groups (G , 〈 ⋅ , ⋅ 〉) with non-Killing left-invariant conformal vector fields. Firstly, we prove that if h is a Cartan subalgebra for a semisimple Levi factor of the Lie algebra g , then dim h ≤ max { 0 , min { p , q } − 2 }. Secondly, we classify trans-Lorentzian Lie groups (i.e., min { p , q } = 2) with non-Killing left-invariant conformal vector fields, and prove that [ g , g ] is at most three-step nilpotent. Thirdly, based on the classification of the trans-Lorentzian Lie groups, we show that the corresponding Ricci operators are nilpotent, and consequently the scalar curvatures vanish. As a byproduct, we prove that four-dimensional trans-Lorentzian Lie groups with non-Killing left-invariant conformal vector fields are necessarily conformally flat, and construct a family of five-dimensional ones which are not conformally flat. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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