{
  "id": "1810.03086",
  "version": "v1",
  "published": "2018-10-07T05:28:59.000Z",
  "updated": "2018-10-07T05:28:59.000Z",
  "title": "Double extensions of restricted Lie algebras",
  "authors": [
    "Said Benayadi",
    "Sofiane Bouarroudj"
  ],
  "comment": "21 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "A double extension ($D$-extension) of a Lie algebra ${\\mathfrak a}$ with a non-degenerate invariant symmetric bilinear form $B_{\\mathfrak a}$, briefly a NIS Lie algebra, is an enlargement of ${\\mathfrak a}$ by means of a central extension and a derivation; the affine Kac-Moody algebras are the most known examples. Let ${\\mathfrak a}$ be a restricted Lie algebra equipped with a NIS $B_{\\mathfrak a}$. Suppose $\\mathfrak a$ has a restricted derivation $D$ such that $B_{\\mathfrak a}$ is $D$-invariant. We show that the double extension of $\\mathfrak a$ caries a $p$-mapping constructed by means of $B_{\\mathfrak a}$ and $D$. We show that, the other way round, any restricted NIS Lie algebra can be obtained as a $D$-extension of another restricted NIS Lie algebra of codimension 2 provided that the center is not trivial together with an extra condition pertaining to the central element. We give examples of $D$-extensions of restricted Lie algebras in small characteristic related with Manin triples for the Heisenberg algebra, and with the Hamiltonian and the Jacobson-Witt algebras. These $D$-extensions are classified up to an isometry; some of them are new.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-07T05:28:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "restricted lie algebra",
      "double extension",
      "restricted nis lie algebra",
      "non-degenerate invariant symmetric bilinear form",
      "affine kac-moody algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}