{
  "abstract": "Let $A$ be a commutative associative unital algebra over a field $K$ of characteristic zero, and let $(G,ρ)$ be a Lie--Rinehart algebra over $A$. We revisit the Gerstenhaber structure carried by the exterior algebra $\\bigwedge_A G$ from the point of view of graded derivations. Particular attention is paid to the well-definedness of the Schouten--Nijenhuis extension over the coefficient algebra $A$: the Lie--Rinehart identity is shown to provide exactly the correction terms needed for the bracket to descend to the $A$-balanced exterior algebra. After establishing graded antisymmetry, the Leibniz rule and the graded Jacobi identity without circularity, we study the inner graded derivations $\\adSN(P)=[P,\\cdot]_{SN}$ and prove \\[ [\\adSN(P),\\adSN(Q)]=\\adSN([P,Q]_{SN}). \\] Finally, we give a complete proof of the converse construction: Gerstenhaber brackets on $\\bigwedge_A G$ are in one-to-one correspondence with Lie--Rinehart structures on $G$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32371",
  "author_total": 1,
  "authors": [
    {
      "name": "Servais Cyr Gatsé \\and Basile Guy Richard Bossoto \\and Irmely Gladesh Mabanza Nsiloulou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/91ps083m9b5fg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32371",
  "fulltext_url": null,
  "id": "91ps083m9b5fg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32371",
  "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.DG"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7214751638",
  "pdf_url": "https://arxiv.org/pdf/2609.32371",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/91ps083m9b5fg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32371",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32371"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Note on the Schouten--Nijenhuis Bracket of Lie--Rinehart Algebras",
  "title_text": "A Note on the Schouten--Nijenhuis Bracket of Lie--Rinehart Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/91ps083m9b5fg",
  "venue": "arXiv",
  "versions": []
}