{
  "id": "0808.1463",
  "version": "v1",
  "published": "2008-08-11T08:22:55.000Z",
  "updated": "2008-08-11T08:22:55.000Z",
  "title": "A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras",
  "authors": [
    "Vyjayanthi Chari",
    "Jacob Greenstein"
  ],
  "comment": "25 pages",
  "journal": "Adv. Math. 220 (2009), no. 4, 1193-1221",
  "doi": "10.1016/j.aim.2008.11.007",
  "categories": [
    "math.RT",
    "math.QA",
    "math.RA"
  ],
  "abstract": "Let $\\lie g$ be a simple Lie algebra and let $\\bs^{\\lie g}$ be the locally finite part of the algebra of invariants $(_\\bc\\bv\\otimes S(\\lie g))^{\\lie g}$ where $\\bv$ is the direct sum of all simple finite-dimensional modules for $\\lie g$ and $S(\\lie g)$ is the symmetric algebra of $\\lie g$. Given an integral weight $\\xi$, let $\\Psi=\\Psi(\\xi)$ be the subset of roots which have maximal scalar product with $\\xi$. Given a dominant integral weight $\\lambda$ and $\\xi$ such that $\\Psi$ is a subset of the positive roots we construct a finite-dimensional subalgebra $\\bs^{\\lie g}_\\Psi(\\le_\\Psi\\lambda)$ of $\\bs^{\\lie g}$ and prove that the algebra is Koszul of global dimension at most the cardinality of $\\Psi$. Using this we then construct naturally an infinite-dimensional Koszul algebra of global dimension equal to the cardinality of $\\Psi$. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-08-11T08:22:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B70",
      "16W50"
    ],
    "keywords": [
      "simple lie algebra",
      "finite-dimensional representations",
      "koszul algebras arising",
      "simple finite-dimensional modules",
      "infinite-dimensional koszul algebra"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0808.1463C"
    }
  }
}