{
  "abstract": "A pre-symplectic left-symmetric algebra $\\mathcal{A}$ is a left-symmetric algebra endowed with a nondegenerate skew-symmetric bilinear form $ω$ such that all left multiplication operators are symmetric with respect to $ω$. In this setting, the underlying subadjacent Lie algebra $(\\mathcal{A}^{-},ω)$ forms a flat $T$-symplectic Lie algebra. This paper provides a systematic investigation into the structural properties of pre-symplectic left-symmetric algebras. In particular, we introduce a distinguished subclass termed \\emph{Milnor pre-symplectic algebras}, and prove that any pre-symplectic left-symmetric algebra whose commutator ideal is nondegenerate necessarily belongs to this subclass. Next, we investigate the Levi-Civita product associated with symplectic Lie algebras. We show that this product always yields a right-symmetric algebra, and we prove that it forms a left-symmetric algebra if and only if it is associative. Furthermore, we provide a characterization of symplectic Lie algebras in terms of representations of left-symmetric algebras, and conclude by establishing a construction method for these structures known as the $T^*$-extension. Furthermore, we develop a double extension procedure for pre-symplectic left-symmetric algebras by means of commutative associative algebras. We show that every such algebra with a degenerate commutator ideal can be reconstructed via this extension process. More generally, we show that any pre-symplectic left-symmetric algebra is either a Milnor pre-symplectic algebra or can be obtained through a finite sequence of successive double extensions starting from a Milnor pre-symplectic algebra. As a concrete application of these structural results, we provide a complete classification of pre-symplectic left-symmetric algebras of dimension less than or equal to $4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.08517",
  "author_total": 2,
  "authors": [
    {
      "name": "Saïd Benayadi"
    },
    {
      "name": "Hamza El Ouali"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ms49fwwmr2q9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.08517",
  "fulltext_url": null,
  "id": "9ms49fwwmr2q9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.08517",
  "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"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7202212166",
  "pdf_url": "https://arxiv.org/pdf/2608.08517",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ms49fwwmr2q9.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:09Z",
      "native_id": "2608.08517",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.08517"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Pre-symplectic left-symmetric algebras",
  "title_text": "Pre-symplectic left-symmetric algebras",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/9ms49fwwmr2q9",
  "venue": "arXiv",
  "versions": []
}