{
  "abstract": "We introduce a homotopical framework for Nijenhuis operators on semistrict Lie $n$-algebras ($n$-term $L_\\infty$-algebras) for every $n \\ge 1$ and, more generally, on arbitrary $L_\\infty$-algebras. A Nijenhuis operator is defined as a family $N = (N_q)_{q \\ge 1}$ of multilinear maps, with $N_1$ linear, realizing an $L_\\infty$-morphism from an $N$-twisted copy of the structure to the original one. We prove that this condition is equivalent to the vanishing, at every arity $m$, of the $m$-th Nijenhuis torsion up to a correction term $Λ^N_m$ formed by higher-component substitutions, and we provide a closed combinatorial formula for $Λ^N_m$ as a sum over set partitions. Along the way, we show that the arity-$k$ Nijenhuis torsion factors canonically as a composite of $k$ commuting elementary slot deformations. The resulting objects form a category $\\mathbf{NijLie}_n$; we establish that $\\mathbf{NijLie}_m$ embeds as the full subcategory of $\\mathbf{NijLie}_n$ on objects concentrated in the bottom $m$ degrees for all $m \\le n$. These levels assemble into a tower whose first three stages are classical Nijenhuis Lie algebras, the Nijenhuis Lie $2$-algebras of Das, and the newly introduced Nijenhuis Lie $3$-algebras. We characterize the strict and skeletal objects at every level, introduce $n$-term representations up to homotopy of a Nijenhuis Lie algebra, and prove that the semidirect product construction establishes an equivalence between the category of such representations and split abelian $\\mathbf{NijLie}_n$-extensions. Finally, two corrections to published level-three identities are documented, and all component identities up to arity five are verified via computer algebra.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37698",
  "author_total": 3,
  "authors": [
    {
      "name": "Imed Basdouri"
    },
    {
      "name": "Tarek Ben Massoud"
    },
    {
      "name": "Ahmed Zahari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/30evk42c0e4kz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37698",
  "fulltext_url": null,
  "id": "30evk42c0e4kz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37698",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37698",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/30evk42c0e4kz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37698",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37698"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Nijenhuis Lie $n$-algebras",
  "title_text": "Nijenhuis Lie n-algebras",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/30evk42c0e4kz",
  "venue": "arXiv",
  "versions": []
}