发表机构
CONICET and Departamento de Matemática, ECEN-FCEIA, Universidad Nacional de Rosario（阿根廷国家科学研究和技术促进委员会和罗萨里奥国立大学经济与工商学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究关于特定表示群、具有自然约化左不变洛伦兹度量的两步幂零李群，通过弱化条件构建框架，涵盖中心非退化和退化情况，得到李代数、表示分解及等距自同构群相关结果，完成结构描述。
AI 中文摘要
我们研究关于表示群\(N \rtimes H^{\operatorname{aut}}\)具有自然约化左不变洛伦兹度量的两步幂零李群。用换位子理想非退化这一较弱条件取代标准的非退化中心假设，我们构建了一个框架，将其构造扩展到洛伦兹情形，涵盖非退化和退化中心情况。在退化情形下，我们表明相关李代数是半直积的中心扩张，其黎曼因子是自然约化的。此外，我们得到定义表示的不变分解，包括一个特殊的洛伦兹因子，并给出等距自同构群的迷向代数和恒等分支的明确描述。这些结果在换位子理想非退化假设下完成了自然约化两步洛伦兹幂零李群的结构描述。
英文摘要
We study $2$-step nilpotent Lie groups with naturally reductive left-invariant Lorentzian metrics with respect to the presentation group $N\rtimes H^{\operatorname{aut}}$. Replacing the standard non-degenerate center assumption with the weaker condition that the commutator ideal be non-degenerate, we develop a framework that extends the representation-theoretic construction to the Lorentzian setting and covers both the non-degenerate and degenerate center cases. In the degenerate case, we show that the associated Lie algebra is a central extension of a semidirect product whose Riemannian factor is naturally reductive. Furthermore, we obtain invariant decompositions of the defining representation in the non-degenerate case. In the degenerate case, we introduce a Riemannian quotient representation and show that the kernel of this representation is a central abelian ideal whose elements act by null rotations. Finally, we use these results to characterize the isotropy algebra of the group of isometric automorphisms.
CommentsThis is a new version of a previously uploaded paper where we correct some minor mistakes and improve some of the results, 31 pages
