{
  "id": "math/0608666",
  "version": "v2",
  "published": "2006-08-28T02:19:21.000Z",
  "updated": "2006-12-05T20:04:35.000Z",
  "title": "Invariant Subspaces of Nilpotent Linear Operators. I",
  "authors": [
    "Claus Michael Ringel",
    "Markus Schmidmeier"
  ],
  "comment": "55 pages, minor modification in (0.1.3), to appear in: Journal fuer die reine und angewandte Mathematik",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $k$ be a field. We consider triples $(V,U,T)$, where $V$ is a finite dimensional $k$-space, $U$ a subspace of $V$ and $T \\:V \\to V$ a linear operator with $T^n = 0$ for some $n$, and such that $T(U) \\subseteq U$. Thus, $T$ is a nilpotent operator on $V$, and $U$ is an invariant subspace with respect to $T$. We will discuss the question whether it is possible to classify these triples. These triples $(V,U,T)$ are the objects of a category with the Krull-Remak-Schmidt property, thus it will be sufficient to deal with indecomposable triples. Obviously, the classification problem depends on $n$, and it will turn out that the decisive case is $n=6.$ For $n < 6$, there are only finitely many isomorphism classes of indecomposables triples, whereas for $n > 6$ we deal with what is called ``wild'' representation type, so no complete classification can be expected. For $n=6$, we will exhibit a complete description of all the indecomposable triples.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-12-05T20:04:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A15",
      "15A21"
    ],
    "keywords": [
      "nilpotent linear operators",
      "invariant subspace",
      "classification problem depends",
      "indecomposable triples",
      "complete classification"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8666R"
    }
  }
}