{
  "id": "2006.15405",
  "version": "v1",
  "published": "2020-06-27T16:55:20.000Z",
  "updated": "2020-06-27T16:55:20.000Z",
  "title": "An algorithm for the periodicity of deformed preprojective algebras of Dynkin types $\\mathbb{E}_6$, $\\mathbb{E}_7$ and $\\mathbb{E}_8$",
  "authors": [
    "Jerzy Białkowski"
  ],
  "comment": "27 pages, 4 figures, 3 algorithms, 5 tables",
  "categories": [
    "math.RT"
  ],
  "abstract": "We construct a numeric algorithm for completing the proof of a conjecture asserting that all deformed preprojective algebras of generalized Dynkin type are periodic. In particular, we obtain an algorithmic procedure showing that non-trivial deformed preprojective algebras of Dynkin types $\\mathbb{E}_7$ and $\\mathbb{E}_8$ exist only in characteristic 2. As a consequence, we show that deformed preprojective algebras of Dynkin types $\\mathbb{E}_6$, $\\mathbb{E}_7$ and $\\mathbb{E}_8$ are periodic and we obtain an algorithm for a classification of such algebras, up to algebra isomorphism. We do it by a reduction of the conjecture to a solution of a system of equations associated with the problem of the existence of a suitable algebra isomorphism $\\varphi_f: P^f(\\mathbb{E}_n) \\to P(\\mathbb{E}_n)$ described in Theorem 2.1. One also shows that our algorithmic approach to the conjecture is also applicable to the classification of the mesh algebras of generalized Dynkin type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-27T16:55:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16D50",
      "16G20",
      "16Z05",
      "65K05",
      "65K10",
      "65H10",
      "65H99",
      "68P05",
      "68W30"
    ],
    "keywords": [
      "generalized dynkin type",
      "periodicity",
      "conjecture",
      "numeric algorithm",
      "algorithmic procedure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}