{
  "id": "1705.07307",
  "version": "v1",
  "published": "2017-05-20T13:56:57.000Z",
  "updated": "2017-05-20T13:56:57.000Z",
  "title": "Lie algebras arising from 1-cyclic perfect complexes",
  "authors": [
    "Shiquan Ruan",
    "Jie Sheng",
    "Haicheng Zhang"
  ],
  "comment": "49 pages",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let $A$ be the path algebra of a Dynkin quiver $Q$ over a finite field, and $\\mathscr{P}$ be the category of projective $A$-modules. Denote by $C^1(\\mathscr{P})$ the category of 1-cyclic complexes over $\\mathscr{P}$, and $\\tilde{\\mathfrak{n}}^+$ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in $C^1(\\mathscr{P})$. In this paper, we prove the existence of Hall polynomials in $C^1(\\mathscr{P})$, and then establish a relationship between the Hall numbers for indecomposable objects therein and those for $A$-modules. Using Hall polynomials evaluated at $1$, we define a Lie bracket in $\\tilde{\\mathfrak{n}}^+$ by the commutators of degenerate Hall multiplication. The resulting Hall Lie algebras provide a broad class of nilpotent Lie algebras. For example, if $Q$ is bipartite, $\\tilde{\\mathfrak{n}}^+$ is isomorphic to the nilpotent part of the corresponding semisimple Lie algebra; if $Q$ is the linearly oriented quiver of type $\\mathbb{A}_{n}$, $\\tilde{\\mathfrak{n}}^+$ is isomorphic to the free 2-step nilpotent Lie algebra with $n$-generators. Furthermore, we give a description of the root systems of different $\\tilde{\\mathfrak{n}}^+$. We also characterize the Lie algebras $\\tilde{\\mathfrak{n}}^+$ by generators and relations. When $Q$ is of type $\\mathbb{A}$, the relations are exactly the defining relations. As a byproduct, we construct an orthogonal exceptional pair satisfying the minimal Horseshoe lemma for each sincere non-projective indecomposable $A$-module.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-20T13:56:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "17B20",
      "17B30"
    ],
    "keywords": [
      "lie algebras arising",
      "perfect complexes",
      "nilpotent lie algebra",
      "hall polynomials",
      "corresponding semisimple lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}