{
  "id": "1109.2915",
  "version": "v1",
  "published": "2011-09-13T20:26:20.000Z",
  "updated": "2011-09-13T20:26:20.000Z",
  "title": "On the invariant theory for tame tilted algebras",
  "authors": [
    "Calin Chindris"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "We show that a tilted algebra $A$ is tame if and only if for each generic root $\\dd$ of $A$ and each indecomposable irreducible component $C$ of $\\module(A,\\dd)$, the field of rational invariants $k(C)^{\\GL(\\dd)}$ is isomorphic to $k$ or $k(x)$. Next, we show that the tame tilted algebras are precisely those tilted algebras $A$ with the property that for each generic root $\\dd$ of $A$ and each indecomposable irreducible component $C \\subseteq \\module(A,\\dd)$, the moduli space $\\M(C)^{ss}_{\\theta}$ is either a point or just $\\mathbb P^1$ whenever $\\theta$ is an integral weight for which $C^s_{\\theta}\\neq \\emptyset$. We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space $\\M(C)^{ss}_{\\theta}$ being smooth for each generic root $\\dd$ of $A$, each indecomposable irreducible component $C \\subseteq \\module(A,\\dd)$, and each integral weight $\\theta$ for which $C^s_{\\theta} \\neq \\emptyset$. As a consequence of this latter description, we show that the smoothness of the various moduli spaces of modules for a strongly simply connected algebra $A$ implies the tameness of $A$. Along the way, we explain how moduli spaces of modules for finite-dimensional algebras behave with respect to tilting functors, and to theta-stable decompositions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-09-13T20:26:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "16G20",
      "16G60",
      "16R30"
    ],
    "keywords": [
      "tame tilted algebras",
      "invariant theory",
      "moduli space",
      "indecomposable irreducible component",
      "generic root"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.2915C"
    }
  }
}