{
  "abstract": "Chu proved that a symplectic structure $ω$ on an even-dimensional Lie algebra induces a left-symmetric structure which is defined by $ω(x\\scriptstyleΔ_ω y,z)=-ω(y,[x,z])$. El Bourkadi and Mansouri proved that a cosymplectic structure on an odd-dimensional Lie algebra $\\mathfrak{g}$ induces the left-symmetric structure, which is defined by using a linear isomorphism from $\\mathfrak{g}$ to $\\mathfrak{g}^*$ associated with the cosymplectic structure. In this paper, for a non-degenerate bilinear form $ϕ$ on a Lie algebra $\\mathfrak{g}$, we give a necessary and sufficient condition for the product $\\scriptstyleΔ_ϕ$ on $\\mathfrak{g}$ defined by $ϕ(x\\scriptstyleΔ_ϕ y,z)=-ϕ(y,[x,z])$ to be a left-symmetric structure. We also prove that the left-symmetric structure $\\scriptstyleΔ_ϕ$ is complete if and only if the Lie algebra $\\mathfrak{g}$ is unimodular. Moreover, we formulate the notion of double extension for non-degenerate bilinear forms and prove that a certain class of non-degenerate bilinear forms is obtained by double extension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33063",
  "author_total": 1,
  "authors": [
    {
      "name": "Naoki Kato"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0x0ef363vsjfj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33063",
  "fulltext_url": null,
  "id": "0x0ef363vsjfj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33063",
  "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.RA"
    ],
    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7214730425",
  "pdf_url": "https://arxiv.org/pdf/2609.33063",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0x0ef363vsjfj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33063",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33063"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Non-degenerate bilinear forms and left-symmetric structures",
  "title_text": "Non-degenerate bilinear forms and left-symmetric structures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/0x0ef363vsjfj",
  "venue": "arXiv",
  "versions": []
}