{
  "id": "1202.6050",
  "version": "v2",
  "published": "2012-02-27T20:46:46.000Z",
  "updated": "2015-04-12T10:15:40.000Z",
  "title": "Tilting modules for the current algebra of a simple Lie algebra",
  "authors": [
    "Matthew Bennett",
    "Vyjayanthi Chari"
  ],
  "journal": "Proceedings of Symposia in Pure Mathematics (86): Recent Developments in Lie Algebras, Groups and Representation Theory 2012, 75-97",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "The category of level zero representations of current and affine Lie algebras shares many of the properties of other well-known categories which appear in Lie theory and in algebraic groups in characteristic p and in this paper we explore further similarities. The role of the standard and co-standard module is played by the finite-dimensional local Weyl module and the dual of the infinite-dimensional global Weyl module respectively. We define the canonical filtration of a graded module for the current algebra. In the case when $\\mathfrak g$ is of type $\\mathfrak{sl}_{n+1}$ we show that the well-known necessary and sufficient homological condition for a canonical filtration to be a good (or a $\\nabla$-filtration) also holds in our situation. Finally, we construct the indecomposable tilting modules in our category and show that any tilting module is isomorphic to a direct sum of indecomposable tilting modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-27T20:46:46.000Z",
      "abstract": "The category of level zero representations of current and affine Lie algebras shares many of the properties of other well-known categories which appear in Lie theory and in algebraic groups in characteristic p and in this paper we explore further similarities. The role of the standard and co-standard module is played by the finite-dimensional local Weyl module and the dual of the infinite-dimensional global Weyl module respectively. We define the canonical filtration of a graded module for the current algebra. In the case when $\\lie g$ is of type $\\lie{sl}_{n+1}$ we show that the well-known necessary and sufficient homological condition for a canonical filtration to be a good (or a $\\nabla$-filtration) also holds in our situation. Finally, we construct the indecomposable tilting modules in our category and show that any tilting module is isomorphic to a direct sum of indecomposable tilting modules.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-12T10:15:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple lie algebra",
      "current algebra",
      "affine lie algebras shares",
      "indecomposable tilting modules",
      "finite-dimensional local weyl module"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Represent. Theory"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.6050B"
    }
  }
}