{
  "id": "2302.01828",
  "version": "v1",
  "published": "2023-02-03T16:04:07.000Z",
  "updated": "2023-02-03T16:04:07.000Z",
  "title": "Exact Borel subalgebras of path algebras of quivers of Dynkin type $\\mathbb{A}$",
  "authors": [
    "Markus Thuresson"
  ],
  "comment": "36 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Hereditary algebras are quasi-hereditary with respect to any adapted partial order on the indexing set of the isomorphism classes of their simple modules. For any adapted partial order on $\\{1,\\dots, n\\}$, we compute the quiver and relations for the $\\operatorname{Ext}$-algebra of standard modules over the path algebra of a uniformly oriented linear quiver with $n$ vertices. Such a path algebra always admits a regular exact Borel subalgebra in the sense of K\\\"onig and we show that there is always a regular exact Borel subalgebra containg the idempotents $e_1,\\dots, e_n$ and find a minimal generating set for it. For a quiver $Q$ and a deconcatenation $Q=Q^1\\sqcup Q^2$ of $Q$ at a sink or source $v$, we describe the $\\operatorname{Ext}$-algebra of standard modules over $KQ$, up to an isomorphism of associative algebras, in terms of that over $KQ^1$ and $KQ^2$. Moreover, we determine necessary and sufficient conditions for $KQ$ to admit a regular exact Borel subalgebra, provided that $KQ^1$ and $KQ^2$ do. We use these results to obtain sufficient and necessary conditions for a path algebra of a linear quiver with arbitrary orientation to admit a regular exact Borel subalgebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-03T16:04:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16D90",
      "16E30"
    ],
    "keywords": [
      "regular exact borel subalgebra",
      "path algebra",
      "dynkin type",
      "exact borel subalgebra containg",
      "adapted partial order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}