{
  "id": "1408.3913",
  "version": "v1",
  "published": "2014-08-18T07:35:59.000Z",
  "updated": "2014-08-18T07:35:59.000Z",
  "title": "Elementary Subalgebrs of Lie Algebras",
  "authors": [
    "Jon F. Carlson",
    "Eric M. Friedlander",
    "Julia Pevtsova"
  ],
  "comment": "Replaces 1st half of arXiv:1207.5898 (with same title). To appear in the Journal of Algebra",
  "categories": [
    "math.RT"
  ],
  "abstract": "We initiate the investigation of the projective varieties $\\mathbb E(r,\\mathfrak g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $\\mathfrak g$ for various $r \\geq 1$. These varieties $\\mathbb E(r,\\mathfrak g)$ are the natural ambient varieties for generalized support varieties for restricted representations of $\\mathfrak g$. We identify these varieties in special cases, revealing their interesting and varied geometric structures. We also introduce invariants for a finite dimensional $\\mathfrak u(\\mathfrak g)$-module $M$, the local $(r,j)$-radical rank and local $(r,j)$-socle rank, functions which are lower/upper semicontinuous on $\\mathbb E(r,\\mathfrak g)$. Examples are given of $\\mathfrak u(\\mathfrak g)$-modules for which some of these rank functions are constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-18T07:35:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B50",
      "16G10"
    ],
    "keywords": [
      "lie algebra",
      "elementary subalgebrs",
      "natural ambient varieties",
      "generalized support varieties",
      "elementary subalgebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3913C"
    }
  }
}