{
  "id": "1105.3407",
  "version": "v2",
  "published": "2011-05-17T15:04:51.000Z",
  "updated": "2012-05-25T11:16:49.000Z",
  "title": "A Koszul category of representations of finitary Lie algebras",
  "authors": [
    "Elizabeth Dan-Cohen",
    "Ivan Penkov",
    "Vera Serganova"
  ],
  "comment": "22 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We find for each simple finitary Lie algebra $\\mathfrak{g}$ a category $\\mathbb{T}_\\mathfrak{g}$ of integrable modules in which the tensor product of copies of the natural and conatural modules are injective. The objects in $\\mathbb{T}_\\mathfrak{g}$ can be defined as the finite length absolute weight modules, where by absolute weight module we mean a module which is a weight module for every splitting Cartan subalgebra of $\\mathfrak{g}$. The category $\\mathbb{T}_\\mathfrak{g}$ is Koszul in the sense that it is antiequivalent to the category of locally unitary finite-dimensional modules over a certain direct limit of finite-dimensional Koszul algebras. We describe these finite-dimensional algebras explicitly. We also prove an equivalence of the categories $\\mathbb{T}_{o(\\infty)}$ and $\\mathbb{T}_{sp(\\infty)}$ corresponding respectively to the orthogonal and symplectic finitary Lie algebras $o(\\infty)$, $sp(\\infty)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-05-25T11:16:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B65",
      "17B10",
      "16G10"
    ],
    "keywords": [
      "koszul category",
      "finite length absolute weight modules",
      "simple finitary lie algebra",
      "symplectic finitary lie algebras",
      "representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.3407D"
    }
  }
}