{
  "id": "1105.2840",
  "version": "v3",
  "published": "2011-05-13T21:22:40.000Z",
  "updated": "2011-10-05T17:55:17.000Z",
  "title": "Faces of polytopes and Koszul algebras",
  "authors": [
    "Vyjayanthi Chari",
    "Apoorva Khare",
    "Tim Ridenour"
  ],
  "comment": "v3: Significant revisions. To appear in the Journal of Pure and Applied Algebra; 20 pages",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let $\\g$ be a reductive Lie algebra and $V$ a $\\g$-semisimple module. In this article, we study the category $\\G$ of graded finite-dimensional representations of $\\g \\ltimes V$. We produce a large class of truncated subcategories, which are directed and highest weight. Suppose $V$ is finite-dimensional with weights $\\wt(V)$. Let $\\Psi \\subset \\wt(V)$ be the set of weights contained in a face $\\F$ of the polytope that is the convex hull of $\\wt(V)$. For each such $\\Psi$, we produce quasi-hereditary Koszul algebras. We use these Koszul algebras to construct an infinite-dimensional graded subalgebra $\\spg$ of the locally finite part of the algebra of invariants $(END{\\C} (\\V) \\otimes \\Sym V)^{\\g}$, where $\\V$ is the direct sum of all simple finite-dimensional $\\g$-modules. We prove that $\\spg$ is Koszul of finite global dimension.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-10-05T17:55:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S37",
      "16W50",
      "17B10"
    ],
    "keywords": [
      "produce quasi-hereditary koszul algebras",
      "finite global dimension",
      "large class",
      "highest weight",
      "reductive lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.2840C"
    }
  }
}