{
  "id": "2209.06258",
  "version": "v1",
  "published": "2022-09-13T18:35:29.000Z",
  "updated": "2022-09-13T18:35:29.000Z",
  "title": "Cluster Nature of Quantum Groups",
  "authors": [
    "Linhui Shen"
  ],
  "comment": "32 pages, 9 figures",
  "categories": [
    "math.RT",
    "math.AG",
    "math.QA"
  ],
  "abstract": "We present a rigid cluster model to realize the quantum group ${\\bf U}_q(\\mathfrak{g})$ for $\\mathfrak{g}$ of type ADE. That is, we prove that there is a natural Hopf algebra isomorphism from the quantum group ${\\bf U}_q(\\mathfrak{g})$ to a quotient algebra of the Weyl group invariants of the Fock-Goncharov quantum cluster algebra $\\mathcal{O}_q(\\mathscr{P}_{{\\rm G},\\odot})$. By applying the quantum duality of cluster algebras, we show that ${\\bf U}_q(\\mathfrak{g})$ admits a natural basis $\\bar{\\bf \\Theta}$ whose structural coefficients are in $\\mathbb{N}[q^{\\frac{1}{2}}, q^{-\\frac{1}{2}}]$. The basis $\\bar{\\bf \\Theta}$ satisfies an invariance property under Lusztig's braid group action, the Dynkin automorphisms, and the star anti-involution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-13T18:35:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B37",
      "13F60",
      "53D30"
    ],
    "keywords": [
      "quantum group",
      "cluster nature",
      "fock-goncharov quantum cluster algebra",
      "lusztigs braid group action",
      "natural hopf algebra isomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}