{
  "id": "2212.02106",
  "version": "v1",
  "published": "2022-12-05T09:03:52.000Z",
  "updated": "2022-12-05T09:03:52.000Z",
  "title": "$U(\\frak h)$-free modules over the Lie algebras of differential operators",
  "authors": [
    "Munayim Dilxat",
    "Shoulan Gao",
    "Dong Liu",
    "Limeng Xia"
  ],
  "comment": "Latex, 17 pages",
  "journal": "Published in Mathematics, 2022, 10(10), 1728",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP",
    "math.RA"
  ],
  "abstract": "In this paper, we consider some non-weight modules over the Lie algebra of Weyl type. First, we determine the modules whose restriction to $U(\\frak h)$ are free of rank $1$ over the Lie algebra of differential operators on the circle. Then we determine the necessary and sufficient conditions for the tensor products of quasi-finite highest weight modules and $U(\\frak h)$-free modules to be irreducible, and obtain that any two such tensor products are isomorphic if and only if the corresponding highest weight modules and $U(\\frak h)$-free modules are isomorphic. Finally, we extend such results to the Lie algebras of differential operators in the general case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-05T09:03:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B65",
      "17B68"
    ],
    "keywords": [
      "lie algebra",
      "free modules",
      "differential operators",
      "quasi-finite highest weight modules",
      "tensor products"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}