{
  "id": "1512.02261",
  "version": "v1",
  "published": "2015-12-07T21:55:53.000Z",
  "updated": "2015-12-07T21:55:53.000Z",
  "title": "Homogeneous Rota-Baxter operators on $3$-Lie algebra $A_ω$",
  "authors": [
    "Ruipu Bai",
    "Yinghua Zhang"
  ],
  "comment": "16pages. arXiv admin note: text overlap with arXiv:1407.3159 by other authors",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the paper we study homogeneous Rota-Baxter operators with weight zero on the infinite dimensional simple $3$-Lie algebra $A_{\\omega}$ over a field $F$ ( $ch F=0$ ) which is realized by an associative commutative algebra $A$ and a derivation $\\Delta$ and an involution $\\omega$ ( Lemma \\mref{lem:rbd3} ). A homogeneous Rota-Baxter operator on $A_{\\omega}$ is a linear map $R$ of $A_{\\omega}$ satisfying $R(L_m)=f(m)L_m$ for all generators of $A_{\\omega}$, where $f : A_{\\omega} \\rightarrow F$. We proved that $R$ is a homogeneous Rota-Baxter operator on $A_{\\omega}$ if and only if $R$ is the one of the five possibilities $R_{0_1}$, $R_{0_2}$,$R_{0_3}$,$R_{0_4}$ and $R_{0_5}$, which are described in Theorem \\mref{thm:thm1}, \\mref{thm:thm4}, \\mref{thm:thm01}, \\mref{thm:thm03} and \\mref{thm:thm04}. By the five homogeneous Rota-Baxter operators $R_{0_i}$, we construct new $3$-Lie algebras $(A, [ , , ]_i)$ for $1\\leq i\\leq 5$, such that $R_{0_i}$ is the homogeneous Rota-Baxter operator on $3$-Lie algebra $(A, [ , , ]_i)$, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-07T21:55:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lie algebra",
      "study homogeneous rota-baxter operators",
      "infinite dimensional simple",
      "weight zero",
      "linear map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}