{
  "id": "1805.04041",
  "version": "v1",
  "published": "2018-05-10T16:15:03.000Z",
  "updated": "2018-05-10T16:15:03.000Z",
  "title": "Skew group algebras of Jacobian algebras",
  "authors": [
    "Simone Giovannini",
    "Andrea Pasquali"
  ],
  "comment": "34 pages, comments welcome",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "For a quiver with potential $(Q,W)$ with an action of a finite cyclic group $G$, we study the skew group algebra $\\Lambda G$ of the Jacobian algebra $\\Lambda = \\mathcal P(Q, W)$. By a result of Reiten and Riedtmann, the quiver $Q_G$ of a basic algebra $\\eta( \\Lambda G) \\eta$ Morita equivalent to $\\Lambda G$ is known. Under some assumptions on the action of $G$, we construct a potential $W_G$ on $Q_G$ such that $\\eta(\\Lambda G) \\eta\\cong \\mathcal P(Q_G , W_G)$. The original quiver with potential can then be recovered by the skew group algebra construction with a natural action of the dual group of $G$. If $\\Lambda$ is self-injective, then $\\Lambda G$ is as well, and we investigate this case. Motivated by Herschend and Iyama's characterisation of 2-representation finite algebras, we study how cuts on $(Q,W)$ behave with respect to our construction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-10T16:15:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16S35"
    ],
    "keywords": [
      "jacobian algebra",
      "skew group algebra construction",
      "finite cyclic group",
      "morita equivalent",
      "finite algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}