{
  "id": "1410.0173",
  "version": "v1",
  "published": "2014-10-01T10:27:29.000Z",
  "updated": "2014-10-01T10:27:29.000Z",
  "title": "The right-hand side of the Jacobi identity: to be naught or not to be?",
  "authors": [
    "Arthemy V. Kiselev"
  ],
  "comment": "Proc. 7th Int. workshop \"Group analysis of differential equations and integrable systems\" (15-19 June 2014, Larnaca, Cyprus); illustration to arXiv:1312.1262 [math-ph] and arXiv:1312.4140 [math-ph]; 14 pages. Main example (sec.2) = [previously unpublished] Appendix in arXiv:1312.4140 (pp. i-iii), main counterexample (sec.4) rephrased from arXiv:1302.4388 (joint with S.Ringers)",
  "categories": [
    "math-ph",
    "math.DG",
    "math.MP",
    "math.SG",
    "nlin.SI"
  ],
  "abstract": "By studying one example of three variational multi-vectors in two very different ways, we inspect the mechanism(s) for validity of Jacobi's identity for the variational Schouten bracket (e.g., in the geometry of Poisson bi-vectors or Batalin-Vilkovisky formalism). Namely, we illustrate and contrast the logics of \"genuine\" and \"naive\" geometries of iterated variations which are contained in the well-known identities satisfied by the Schouten bracket and BV-Laplacian. Whereas the true picture from [arXiv:1312.1262] and [arXiv:1312.4140] keeps track of several copies of the integration manifold in both the functionals and pairs of variations' parity-even and odd components, too early does the traditional approach merge the integration domains. Using an elementary counterexample, we point at an inconsistency in the traditional paradigm. PACS: 02.40.-k, 11.10.-z; also 02.30.Ik, 02.30.Jr, 11.15.-q, 11.30.-j",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-01T10:27:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37K10",
      "53D17",
      "70S05",
      "35A15",
      "37K25",
      "81S10"
    ],
    "keywords": [
      "jacobi identity",
      "right-hand side",
      "variational schouten bracket",
      "traditional approach merge",
      "traditional paradigm"
    ],
    "publication": {
      "doi": "10.1088/1742-6596/670/1/012030",
      "journal": "Journal of Physics Conference Series",
      "year": 2016,
      "month": "Jan",
      "volume": 670,
      "number": 1,
      "pages": "012030"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1319681,
      "adsabs": "2016JPhCS.670a2030K"
    }
  }
}