{
  "abstract": "We study local and $2$-local derivations of finite-dimensional Zinbiel algebras. Over any field of characteristic zero, every Zinbiel algebra $A$ with $A^2 \\neq 0$ and $\\dim A \\leq 4$ admits a proper local derivation. The proof uses a filtration argument. We exhibit two five-dimensional Zinbiel algebras with nonzero multiplication whose derivations are all central. Every local derivation on either algebra is therefore a derivation, so the minimum dimension for local rigidity with nonzero multiplication is five. Annihilator-valued interpolation gives a general construction of proper local derivations. When $A^3 = 0$, the multiplication tensor describes all derivations and local derivations. For a finite-dimensional nilpotent algebra $A$ over any field, we show that every $2$-local derivation is a derivation if and only if $\\dim(A/A^2) \\leq 1$. The proof combines the annihilator interpolation of Sh. Ayupov, K. Kudaybergenov and B. Omirov with a one-generator argument. When all derivations are central, every $2$-local derivation is obtained by composing a homogeneous map $A/A^2 \\to \\text{Ann}(A)$ with the quotient map, and every such composition is a $2$-local derivation. The five-dimensional examples extend to locally rigid Zinbiel algebras with nonlinear $2$-local derivations in every dimension at least five. We also show that derivations given by left multiplication form a Lie ideal of nilpotency class at most two and give sufficient conditions for the existence of left outer derivations. This ideal is zero in our locally rigid examples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07863",
  "author_total": 3,
  "authors": [
    {
      "name": "Arislanbay Abdireymov"
    },
    {
      "name": "Aloberdi Sattarov"
    },
    {
      "name": "Bakhtiyor Yusupov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9kf5xwrkcqw9t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07863",
  "fulltext_url": null,
  "id": "9kf5xwrkcqw9t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07863",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07863",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9kf5xwrkcqw9t.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07863",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07863"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Local and 2-Local Derivations of Finite-Dimensional Zinbiel Algebras",
  "title_text": "Local and 2-Local Derivations of Finite-Dimensional Zinbiel Algebras",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/9kf5xwrkcqw9t",
  "venue": "arXiv",
  "versions": []
}