{
  "id": "1504.03863",
  "version": "v1",
  "published": "2015-04-15T11:32:03.000Z",
  "updated": "2015-04-15T11:32:03.000Z",
  "title": "New realization of cyclotomic $q$-Schur algebras I",
  "authors": [
    "Kentaro Wada"
  ],
  "comment": "58 pages",
  "categories": [
    "math.RT",
    "math.CO",
    "math.QA"
  ],
  "abstract": "We introduce a Lie algebra $\\mathfrak{g}_{\\mathbf{Q}}(\\mathbf{m})$ and an associative algebra $\\mathcal{U}_{q,\\mathbf{Q}}(\\mathbf{m})$ associated with the Cartan data of $\\mathfrak{gl}_m$ which is separated into $r$ parts with respect to $\\mathbf{m}=(m_1, \\dots, m_r)$ such that $m_1+ \\dots + m_r =m$. We show that the Lie algebra $\\mathfrak{g}_{\\mathbf{Q}} (\\mathbf{m})$ is a filtered deformation of the current Lie algebra of $\\mathfrak{gl}_m$, and we can regard the algebra $\\mathcal{U}_{q, \\mathbf{Q}}(\\mathbf{m})$ as a \"$q$-analogue\" of $U(\\mathfrak{g}_{\\mathbf{Q}}(\\mathbf{m}))$. Then, we realize a cyclotomic $q$-Schur algebra as a quotient algebra of $\\mathcal{U}_{q, \\mathbf{Q}}(\\mathbf{m})$ under a certain mild condition. We also study the representation theory for $\\mathfrak{g}_{\\mathbf{Q}}(\\mathbf{m})$ and $\\mathcal{U}_{q,\\mathbf{Q}}(\\mathbf{m})$, and we apply them to the representations of the cyclotomic $q$-Schur algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-15T11:32:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G43",
      "17B10",
      "17B37"
    ],
    "keywords": [
      "schur algebra",
      "cyclotomic",
      "realization",
      "current lie algebra",
      "representation theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150403863W"
    }
  }
}