{
  "abstract": "Let $n\\geq 3$, let $H\\subseteq\\mathbb{C}^n$ be any additive subgroup spanning $\\mathbb{C}^n$, and let $0\\ne t\\in H$. We study Pozhidaev's central simple $n$-Lie algebra $P(H,t)=\\widetilde{\\mathcal A}(H,t)/\\mathbb{C}e_0$ without a finite generation or discreteness assumption on $H$. Its inner derivation algebra is the simple generalized divergence-free Lie algebra $\\mathcal S(0,0,n;t,H)$. We prove that its space of inner-equivariant symmetric products vanishes and that every $1/n$-derivation is a scalar multiple of the identity. Using these invariants, we show that $P(H,t)$ is not isomorphic to any simple nonabelian $n$-Lie algebra defined on the underlying spaces of the $S$, $W$, or $SW$ constructions recorded by Cantarini and Kac. We also show that $P(H,t)$ is the quotient by the constants of the derived algebra of an explicit $S$-algebra on $\\mathbb{C}[H]$. Finally, we realize Pozhidaev's second construction $E(H)$ over $\\mathbb{C}$ as a $W$-algebra.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07569",
  "author_total": 3,
  "authors": [
    {
      "name": "Xinru Cao"
    },
    {
      "name": "Bakhrom A. Omirov"
    },
    {
      "name": "Yuhui Tan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1327svt1exfdk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07569",
  "fulltext_url": null,
  "id": "1327svt1exfdk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07569",
  "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": "W7211992378",
  "pdf_url": "https://arxiv.org/pdf/2609.07569",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1327svt1exfdk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07569",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07569"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Pozhidaev and Cantarini--Kac Constructions of Simple $n$-Lie Algebras: Distinctions and Realizations",
  "title_text": "The Pozhidaev and Cantarini--Kac Constructions of Simple n-Lie Algebras: Distinctions and Realizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/1327svt1exfdk",
  "venue": "arXiv",
  "versions": []
}