{
  "id": "2006.07285",
  "version": "v1",
  "published": "2020-06-12T16:09:41.000Z",
  "updated": "2020-06-12T16:09:41.000Z",
  "title": "Completions of discrete cluster categories of type $\\mathbb{A}$",
  "authors": [
    "Charles Paquette",
    "Emine Yildirim"
  ],
  "comment": "30 pages, 10 figures",
  "categories": [
    "math.RT"
  ],
  "abstract": "We complete the discrete cluster categories of type $\\mathbb{A}$ as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then take a certain Verdier quotient. The resulting category is a Hom-finite Krull-Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom-spaces in this new category can be described geometrically, even though the category is not $2$-Calabi-Yau and Ext-spaces are not always symmetric. We describe all cluster-tilting subcategories. Given such a subcategory, we define a cluster character that takes value in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite dimensional sub-representations of infinite dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi-Yau conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-12T16:09:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18E30",
      "16G20"
    ],
    "keywords": [
      "completions",
      "discrete cluster category inside",
      "account infinite dimensional sub-representations",
      "cluster character",
      "subcategory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}