{
  "id": "1911.03316",
  "version": "v1",
  "published": "2019-11-08T15:15:42.000Z",
  "updated": "2019-11-08T15:15:42.000Z",
  "title": "Branching problem on winding subalgebras of affine Kac-Moody algebras A^{(1)}_1 and A^{(2)}_2",
  "authors": [
    "Khanh Nguyen Duc"
  ],
  "comment": "29 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Consider an affine Kac-Moody algebra $\\mathfrak{g}$ with Cartan subalgebra $\\lh$. Given $\\Lambda$ in the set $P_+$ of dominant integral weights of $\\mathfrak{g}$, we denote by $L(\\Lambda)$ the integrable highest weight $\\mathfrak{g}$-module with highest weight $\\Lambda$. For $\\mu\\in \\lh^*$, we denote by $L(\\Lambda)_\\mu$ the corresponding weight space. Consider the support $\\Gamma(\\lg,\\lh)$ of the decompositions of the $L(\\Lambda)$ as a $\\lh$-module: $$ \\Gamma(\\lg,\\lh)=\\{(\\Lambda,\\mu)\\,:\\, L(\\Lambda)_\\mu\\neq\\{0\\}\\}. $$ Consider now the winding subalgebra $\\mathfrak{g}[u]$ (for some positive integer $u$). The winding subalgebra $\\mathfrak{g}[u]$ is isomorphic to $\\lg$ but with a nontrivial embedding in $\\lg$ depending on the parameter $u$. Given $\\lambda$ in the set $\\dot{P}_+$ of dominant integral weights of $\\mathfrak{g}[u]$, we denote by $\\dot{L}(\\lambda)$ the integrable highest weight $\\mathfrak{g}[u]$-module with highest weight $\\lambda$. Then the $\\mathfrak{g}$-module $L(\\Lambda)$ decomposes as a direct sums of simple $\\lg[u]$-modules $\\dot{L}(\\lambda)$ with finite multiplicities. In this paper, we are interested in the supports of this decomposition, i.e., the set of pairs $(\\Lambda,\\lambda)$ in $P_+ \\times \\dot{P}_+$ such that the integrable highest weight $\\mathfrak{g}[u]$-modules $\\dot{L}(\\lambda)$ is a submodule of $L(\\Lambda)$. We show that both $\\Gamma(\\lg,\\lh)$ and $\\Gamma(\\lg,\\lg[u])$ are semigroups. Moreover, for the cases $A^{(1)}_1$ and $A^{(2)}_2$, we determine explicitly $\\Gamma(\\lg,\\lh)$. Finaly, we describe explicit subsets of $P_+ \\times \\dot{P}_+$ where the two semigroups coincide.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-08T15:15:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B67",
      "17B10",
      "22E65"
    ],
    "keywords": [
      "affine kac-moody algebra",
      "winding subalgebra",
      "integrable highest weight",
      "branching problem",
      "dominant integral weights"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}