{
  "id": "1210.3785",
  "version": "v1",
  "published": "2012-10-14T12:15:51.000Z",
  "updated": "2012-10-14T12:15:51.000Z",
  "title": "Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "28 pages",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "We study certain \"\\sigma-commuting varieties\" associated with a pair of commuting involutions of a semisimple Lie algebra $\\g$. The usual commuting variety of $\\g$ and commuting varieties related to one involution are particular cases of our construction. We develop a general theory of \\sigma-commuting varieties and point out some cases, when they have especially good properties. We show that, for a special choice of commuting involutions, the \\sigma-commuting variety is isomorphic to the commuting variety of a simple Jordan algebra. As a by-product of our theory, we show that if $J$ is the Jordan algebra of symmetric matrices, then the product map $J \\times J\\to J$ is equidimensional; while for all other simple Jordan algebras equidimensionality fails.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-14T12:15:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30",
      "17B08",
      "17B40",
      "17C20",
      "22E46"
    ],
    "keywords": [
      "commuting variety",
      "commuting involutions",
      "simple jordan algebras equidimensionality fails",
      "semisimple lie algebra",
      "general theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.3785P"
    }
  }
}