{
  "id": "math/0403203",
  "version": "v2",
  "published": "2004-03-12T00:32:56.000Z",
  "updated": "2005-08-23T22:34:20.000Z",
  "title": "Representation rings of Lie superalgebras",
  "authors": [
    "Gregory D. Landweber"
  ],
  "comment": "36 pages, 1 figure, uses Payl Taylor's diagrams package. Updated with minor corrections",
  "journal": "K-Theory 36 (2005), no. 1-2, 115-168",
  "categories": [
    "math.RT",
    "math.KT"
  ],
  "abstract": "Given a Lie superalgebra \\g, we introduce several variants of the representation ring, built as subrings and quotients of the ring R_{\\Z_2}(\\g) of virtual \\g-supermodules (up to even isomorphisms). In particular, we consider the ideal R_{+}(\\g) of virtual \\g-supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring SR(\\g) on which the parity reversal operator takes the class of a virtual \\g-supermodule to its negative. We also construct representation groups built from ungraded \\g-modules, as well as degree-shifted representation groups using Clifford modules. The full super representation ring SR^{*}(\\g), including all degree shifts, is then a \\Z_{2}-graded ring in the complex case and a \\Z_{8}-graded ring in the real case. Our primary result is a six-term periodic exact sequence relating the rings R^{*}_{\\Z_2}(\\g), R^{*}_{+}(\\g), and SR^{*}(\\g). We first establish a version of it working over an arbitrary (not necessarily algebraically closed) field of characteristic 0. In the complex case, this six-term periodic long exact sequence splits into two three-term sequences, which gives us additional insight into the structure of the complex super representation ring SR^{*}(\\g). In the real case, we obtain the expected 24-term version, as well as a surprising six-term version of this periodic exact sequence.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-08-23T22:34:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "19A22",
      "19L47",
      "17B10",
      "16E20"
    ],
    "keywords": [
      "lie superalgebra",
      "periodic exact sequence relating",
      "k-theoretic super representation ring",
      "periodic long exact sequence splits",
      "six-term periodic long exact sequence"
    ],
    "tags": [
      "research tool",
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......3203L"
    }
  }
}