{
  "id": "2002.05686",
  "version": "v1",
  "published": "2020-02-13T18:08:10.000Z",
  "updated": "2020-02-13T18:08:10.000Z",
  "title": "Tensor categories of affine Lie algebras beyond admissible level",
  "authors": [
    "Thomas Creutzig",
    "Jinwei Yang"
  ],
  "comment": "Comments are welcome",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "We show that if $V$ is a vertex operator algebra of positive energy such that all the irreducible ordinary $V$-modules are $C_1$-cofinite and all the grading-restricted generalized Verma modules for $V$ are of finite length, then the category of finite length generalized $V$-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra associated to $\\mathfrak{g}$ at level $k$ and the category $KL_k(\\mathfrak{g})$ of its ordinary modules, we discover several families of $KL_k(\\mathfrak{g})$ at non-admissible levels $k$, having braided tensor category structures. In particular, $KL_k(\\mathfrak{g})$ has a braided tensor category structure if it is semisimple or more generally if each object in $KL_k(\\mathfrak{g})$ is of finite length. We also prove the rigidity and determine the fusion rules of some categories $KL_k(\\mathfrak{g})$, including the category $KL_{-1}(\\mathfrak{sl}_n)$. Using these results, we construct a rigid tensor category structure on $KL_1(\\mathfrak{sl}(n|m))$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-13T18:08:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B65",
      "17B69",
      "18D10"
    ],
    "keywords": [
      "affine lie algebras",
      "braided tensor category structure",
      "admissible level",
      "finite length",
      "simple affine vertex operator algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}