{
  "id": "1601.02208",
  "version": "v1",
  "published": "2016-01-10T12:29:13.000Z",
  "updated": "2016-01-10T12:29:13.000Z",
  "title": "On axioms of Frobenius like structure in the theory of arrangements",
  "authors": [
    "Alexander Varchenko"
  ],
  "comment": "Latex, 12 pages",
  "categories": [
    "math.AG",
    "nlin.SI"
  ],
  "abstract": "A Frobenius manifold is a manifold with a flat metric and a Frobenius algebra structure on tangent spaces at points of the manifold such that the structure constants of multiplication are given by third derivatives of a potential function on the manifold with respect to flat coordinates. In this paper we present a modification of that notion coming from the theory of arrangements of hyperplanes. Namely, given natural numbers $n>k$, we have a flat $n$-dimensional manifold and a vector space $V$ with a nondegenerate symmetric bilinear form and an algebra structure on $V$, depending on points of the manifold, such that the structure constants of multiplication are given by $2k+1$-st derivatives of a potential function on the manifold with respect to flat coordinates. We call such a structure a {\\it Frobenius like structure}. Such a structure arises when one has a family of arrangements of $n$ affine hyperplanes in $\\C^k$ depending on parameters so that the hyperplanes move parallely to themselves when the parameters change. In that case a Frobenius like structure arises on the base $\\C^n$ of the family.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-10T12:29:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arrangements",
      "structure arises",
      "nondegenerate symmetric bilinear form",
      "flat coordinates",
      "potential function"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160102208V"
    }
  }
}