{
  "id": "1404.1149",
  "version": "v2",
  "published": "2014-04-04T03:59:58.000Z",
  "updated": "2017-10-16T16:23:49.000Z",
  "title": "Generic property and conjugacy classes of homogeneous Borel subalgebras of restricted Lie algebras",
  "authors": [
    "Bin Shu"
  ],
  "comment": "24 pages. The title is changed. In the revised version, we limit to classify the conjugacy classes of homogeneous Borel subalgebras",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $(\\mathfrak{g},[p])$ be a finite-dimensional restricted Lie algebra over an algebraically closed field $\\mathbb{K}$ of characteristic $p>0$, and $G$ be the adjoint group of $\\mathfrak{g}$. We say that $\\mathfrak{g}$ satisfying the {\\sl generic property} if $\\mathfrak{g}$ admits generic tori introduced in \\cite{BFS}. A Borel subalgebra (or Borel for short) of $\\mathfrak{g}$ is by definition a maximal solvable subalgebra containing a maximal torus of $\\mathfrak{g}$, which is further called generic if additionally containing a generic torus. In this paper, we first settle a conjecture proposed by Premet in \\cite{Pr2} on regular Cartan subalgebras of restricted Lie algebras. We prove that the statement in the conjecture for a given $\\mathfrak{g}$ is valid if and only if it is the case when $\\mathfrak{g}$ satisfies the generic property. We then classify the conjugay classes of homogeneous Borel subalgebras of the restricted simple Lie algebras $\\mathfrak{g}=W(n)$ under $G$-conjugation when $p>3$, and present the representatives of these classes. Here $W(n)$ is the so-called Jacobson-Witt algebra, by definition the derivation algebra of the truncated polynomial ring $\\mathbb{K}[T_1,\\cdots,T_n]\\slash (T_1^p,\\cdots,T_n^p)$. We also describe the closed connected solvable subgroups of $G$ associated with those representative Borel subalgebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-04T03:59:58.000Z",
      "title": "Conjugacy of Borel subalgebras of restricted Lie algeberas and the associated solvabel algebraic groups (I)",
      "abstract": "Let (g,[p]) be a finite-dimensional restricted Lie algebra defined over an algebraically closed field K of characteristic p > 0, and G be the adjoint group of g. A Borel subalgebra (or Borel for short) of g is defined as a maximal solvable subalgebra containing a maximal torus of g. Generic Borel subalgebras are by definition a class of Borel subalgebras containing so-called generic Cartan subalgebras. In this paper, we first prove that a conjecture Premet proposed in [17] on regular Cartan subalgebras is valid if and only if it is the case when g has generic Cartan subalgebras. We further prove that all maximal solvable subalgebras of g are Borels whenever p > dimg. We finally classify the conjugay classes of Borel subalgebras of the restricted simple Lie algebras W(n) under G-conjugation when p > 3, and present the representatives of these classes. We also describe the closed connected solvable subgroups of G associated with those representative Borel subalgebras.",
      "comment": "27 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2017-10-16T16:23:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "borel subalgebra",
      "associated solvabel algebraic groups",
      "restricted lie algeberas",
      "generic cartan subalgebras",
      "restricted lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.1149S"
    }
  }
}