{
  "id": "1010.3992",
  "version": "v4",
  "published": "2010-10-19T17:39:20.000Z",
  "updated": "2012-07-26T09:23:02.000Z",
  "title": "Flag manifolds, symmetric $\\fr{t}$-triples and Einstein metrics",
  "authors": [
    "Ioannis Chrysikos"
  ],
  "comment": "18 pages (the text has been reduced to 18 pages, some misprints has been corrected, unchanged results)",
  "categories": [
    "math.DG",
    "math.RT"
  ],
  "abstract": "Let $G$ be a compact connected simple Lie group and let $M=G^{\\bb{C}}/P=G/K$ be a generalized flag manifold. In this article we focus on an important invariant of $G/K$, the so called $\\fr{t}$-root system $R_{\\fr{t}}$, and we introduce the notion of symmetric $\\fr{t}$-triples, that is triples of $\\fr{t}$-roots $\\xi, \\zeta, \\eta\\in R_{\\fr{t}}$ such that $\\xi+\\eta+\\zeta=0$. We describe their properties and we present an interesting application on the structure constants of $G/K$, quantities which are straightforward related to the construction of the homogeneous Einstein metric on $G/K$. Next we classify symmetric $\\fr{t}$-triples for generalized flag manifolds $G/K$ with second Betti number $b_{2}(G/K)=1$, and we treat also the case of full flag manifolds $G/T$, where $T$ is a maximal torus of $G$. In the last section we construct the homogeneous Einstein equation on flag manifolds $G/K$ with five isotropy summands, determined by the simple Lie group $G=\\SO(7)$. By solving the corresponding algebraic system we classify all $\\SO(7)$-invariant (non-isometric) Einstein metrics, and these are the very first results towards the classification of homogeneous Einstein metrics on flag manifolds with five isotropy summands.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2012-07-26T09:23:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C25",
      "53C30"
    ],
    "keywords": [
      "generalized flag manifold",
      "homogeneous einstein metric",
      "isotropy summands",
      "compact connected simple lie group",
      "second betti number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.3992C"
    }
  }
}