{
  "id": "1411.3221",
  "version": "v1",
  "published": "2014-11-12T16:14:51.000Z",
  "updated": "2014-11-12T16:14:51.000Z",
  "title": "Representation embeddings, interpretation functors and controlled wild algebras",
  "authors": [
    "Lorna Gregory",
    "Mike Prest"
  ],
  "categories": [
    "math.RT",
    "math.LO",
    "math.RA"
  ],
  "abstract": "We show that representation embeddings between categories of representations of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of representations of any wild finite-dimensional algebra has width $\\infty$ and hence, if the algebra is countable, there a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: that a representation embedding from ${\\rm Mod}{-}S$ to ${\\rm Mod}{-}R$ admits an inverse interpretation functor from its image and hence that, in this case, ${\\rm Mod}{-}R$ interprets ${\\rm Mod}{-}S$. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-12T16:14:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G60",
      "03C60",
      "03D35",
      "16G20",
      "16D90"
    ],
    "keywords": [
      "representation embedding",
      "controlled wild algebras",
      "finite-dimensional algebras induce embeddings",
      "wild finite-dimensional algebra",
      "inverse interpretation functor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.3221G"
    }
  }
}