{
  "id": "2402.09347",
  "version": "v2",
  "published": "2024-02-14T17:50:30.000Z",
  "updated": "2024-04-06T18:28:13.000Z",
  "title": "Irreducible representations of the crystallisation of the $C^{*}$-algebra $C(SU_{q}(n+1))$",
  "authors": [
    "Manabendra Giri",
    "Arup Kumar Pal"
  ],
  "comment": "v1: 42 pages, 6 diagrams v2: 45 pages, 6 diagrams (first section expanded; some references added)",
  "categories": [
    "math.OA",
    "math.QA"
  ],
  "abstract": "Crystallization of the $C^*$-algebras $C(SU_{q}(n+1))$ was introduced by Giri \\& Pal in arXiv:2203.14665 [math.QA] as a $C^*$-algebra $A_{n}(0)$ given by a finite set of generators and relations. Here we study the irreducible representations of the $C^*$-algebra $A_{n}(0)$ and prove a factorization theorem for its irreducible representations. This leads to a complete classification of all irreducible representations of $A_{n}(0)$. As an important consequence, we prove that all the irreducible representations arise exactly as $q\\to 0+$ limits of the irreducible representations of $C (SU_{q}(n+1))$ given by a result of Soibelman. As a consequence, we show that $A_{n}(0)$ is a type I $C^{*}$-algebra.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-04-06T18:28:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G42",
      "46L67",
      "58B32"
    ],
    "keywords": [
      "crystallisation",
      "finite set",
      "complete classification",
      "important consequence",
      "irreducible representations arise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}