{
  "abstract": "This article provides a complete algebraic description of $\\frac{1}{2}$-derivations, local $\\frac{1}{2}$-derivations, and 2-local $\\frac{1}{2}$-derivations on $n$-dimensional totally graded complex filiform Lie algebras of maximum length. Based on the foundational classification framework established by Janez Bernik (2020), we systematically determine the vector spaces of $\\frac{1}{2}$-derivations for the six infinite structural sequences ($m_0(n)$, $m_2(n)$, $W^+(n)$, $m_{0,1}(n)$, $m_{0,2}(n)$, $m_{0,3}(n)$) and the five exceptional one-parameter families ($g_{7,α}$ through $g_{11,α}$). By analyzing the pointwise local evaluation equations via parametric matrix systems, we establish the structural linearity and rigidity of local $\\frac{1}{2}$-derivations. In contrast, we demonstrate that the independent parameters residing in the boundary rows of the $\\frac{1}{2}$-derivation matrices provide sufficient degrees of freedom to bypass linearity constraints. Exploiting these boundary configurations, we explicitly construct pure non-linear and non-additive 2-local $\\frac{1}{2}$-derivations leveraging the homogeneous function of degree one, $f(z_1, z_2) = z_1^3 / (z_1^2 + z_2^2)$, thereby defining the exact boundary where local rigidity fails.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01556",
  "author_total": 2,
  "authors": [
    {
      "name": "Farkhodzhon Arzikulov"
    },
    {
      "name": "Mirzobek Shodiev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0a1nemv6hxqxk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01556",
  "fulltext_url": null,
  "id": "0a1nemv6hxqxk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01556",
  "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": "W7216467272",
  "pdf_url": "https://arxiv.org/pdf/2610.01556",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0a1nemv6hxqxk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01556",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01556"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Local and 2 local $\\frac{1}{2}$-derivation of $n$-dimensional totally graded filiform Lie algebras",
  "title_text": "Local and 2 local 1/2-derivation of n-dimensional totally graded filiform Lie algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/0a1nemv6hxqxk",
  "venue": "arXiv",
  "versions": []
}