{
  "id": "1304.1461",
  "version": "v3",
  "published": "2013-04-04T18:43:25.000Z",
  "updated": "2013-12-17T16:13:29.000Z",
  "title": "New graded methods in the homological algebra of semisimple algebraic groups",
  "authors": [
    "Brian J. Parshall",
    "Leonard L. Scott"
  ],
  "comment": "Numerous typos and other corrections have been made",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of positive characteristic $p$. Under some restrictions on the size of $p$, the present paper establishes new results on the $G$-module structure of $\\Ext^\\bullet_{G_1}(V,W)$ when $V,W$ belong to several important classes of rational $G$-modules, and $G_1$ denotes the first Frobenius kernel of $G$. For example, it is proved that, if $L,L'$ are ($p$-regular) irreducible $G_1$-modules, then $\\Ext^n_{G_1}(L,L')^{[-1]}$ has a good filtration with computable multiplicities. This and many other results depend on the entirely new technique of using methods of what we call forced gradings in the representation theory of $G$, as developed by the authors in recent papers, and extended here. In addition to providing proofs, these methods lead effectively to a new conceptual framework for the study of rational $G$-modules, and, in this context, to the introduction of a new class of graded finite dimensional algebras, which we call Q-Koszul algebras. These algebras are similar to Koszul algebras, but are quasi-hereditary, rather than semisimple, in grade 0.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-12-17T16:13:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16P10"
    ],
    "keywords": [
      "semisimple algebraic group",
      "graded methods",
      "homological algebra",
      "first frobenius kernel",
      "graded finite dimensional algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.1461P"
    }
  }
}