{
  "id": "1408.4763",
  "version": "v1",
  "published": "2014-08-19T02:10:08.000Z",
  "updated": "2014-08-19T02:10:08.000Z",
  "title": "Symplectic structures on $3$-Lie algebras",
  "authors": [
    "Ruipu Bai",
    "Shuangshuang Chen",
    "Rong Cheng"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:math/0603066 by other authors",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The symplectic structures on $3$-Lie algebras and metric symplectic $3$-Lie algebras are studied. For arbitrary $3$-Lie algebra $L$, infinite many metric symplectic $3$-Lie algebras are constructed. It is proved that a metric $3$-Lie algebra $(A, B)$ is a metric symplectic $3$-Lie algebra if and only if there exists an invertible derivation $D$ such that $D\\in Der_B(A)$, and is also proved that every metric symplectic $3$-Lie algebra $(\\tilde{A}, \\tilde{B}, \\tilde{\\omega})$ is a $T^*_{\\theta}$-extension of a metric symplectic $3$-Lie algebra $(A, B, \\omega)$. Finally, we construct a metric symplectic double extension of a metric symplectic $3$-Lie algebra by means of a special derivation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-19T02:10:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lie algebra",
      "symplectic structures",
      "metric symplectic double extension",
      "special derivation",
      "invertible derivation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1311884,
      "adsabs": "2014arXiv1408.4763B"
    }
  }
}