{
  "id": "1805.12180",
  "version": "v1",
  "published": "2018-05-30T19:04:24.000Z",
  "updated": "2018-05-30T19:04:24.000Z",
  "title": "Gluing of $n$-cluster tilting subcategories for representation-directed algebras",
  "authors": [
    "Laertis Vaso"
  ],
  "comment": "48 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Given $n<d<\\infty$, we investigate the existence of algebras of global dimension $d$ which admit an $n$-cluster tilting subcategory. We construct many such examples using representation-directed algebras. First, given two representation-directed algebras $A$ and $B$, a projective $A$-module $P$ and an injective $B$-module $I$ satisfying certain conditions, we show how we can construct a new representation-directed algebra $\\Lambda$ in such a way that the representation theory of $\\Lambda$ is completely described by the representation theories of $A$ and $B$. Next we introduce $n$-fractured subcategories which generalize $n$-cluster tilting subcategories for representation-directed algebras. We then show how one can construct an $n$-cluster tilting subcategory for $\\Lambda$ by using $n$-fractured subcategories of $A$ and $B$. As an application of our construction, we show that if $n$ is odd and $d\\geq n$ then there exists an algebra admitting an $n$-cluster tilting subcategory and having global dimension $d$. We show the same result if $n$ is even and $d$ is odd or $d\\geq 2n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-30T19:04:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cluster tilting subcategory",
      "representation-directed algebra",
      "global dimension",
      "representation theory",
      "fractured subcategories"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}