{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.18017",
  "author_total": 3,
  "authors": [
    {
      "name": "Brian Luporini"
    },
    {
      "name": "Silvio Reggiani"
    },
    {
      "name": "Francisco Vittone"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a95j1kvxd2871.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.18017",
  "fulltext_url": null,
  "id": "a95j1kvxd2871",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.18017",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2607.18017",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a95j1kvxd2871.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2607.18017",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.18017"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the structure of $2$-step nilpotent Lorentzian naturally reductive Lie groups",
  "title_text": "On the structure of $2$-step nilpotent Lorentzian naturally reductive Lie groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/a95j1kvxd2871",
  "venue": "arXiv",
  "versions": []
}