{
  "id": "0808.2251",
  "version": "v3",
  "published": "2008-08-18T13:21:26.000Z",
  "updated": "2011-09-14T22:15:45.000Z",
  "title": "Compact symmetric spaces, triangular factorization, and Cayley coordinates",
  "authors": [
    "Derek Habermas"
  ],
  "comment": "19 pages: Main proof entirely rewritten, sections reorganized, exposition made more precise and concise. To appear in Pacific Journal of Mathematics",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let U/K represent a connected, compact symmetric space, where theta is an involution of U that fixes K, phi: U/K to U is the geodesic Cartan embedding, and G is the complexification of U. We investigate the intersection of phi(U/K) with the Bruhat decomposition of G corresponding to a theta-stable triangular, or LDU, factorization of the Lie algebra of G. When g in phi(U/K) is generic, the corresponding factorization g=ld(g)u is unique, where l in N^-, d(g) in H, and u in N^+. In this paper we present an explicit formula for d in Cayley coordinates, compute it in several types of symmetric spaces, and use it to identify representatives of the connected components of the generic part of phi(U/K). This formula calculates a moment map for a torus action on the highest dimensional symplectic leaves of the Evens-Lu Poisson structure on U/K.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-09-14T22:15:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C35",
      "43A85",
      "57T15",
      "53D17"
    ],
    "keywords": [
      "compact symmetric space",
      "cayley coordinates",
      "triangular factorization",
      "highest dimensional symplectic leaves",
      "evens-lu poisson structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0808.2251H"
    }
  }
}