{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a connected real or complex [[Lie group|7JBRGCVJRB]], and let $\\mathfrak{g}$ be its Lie algebra.\nThe group $G$ is a *simple Lie group* if $\\mathfrak{g}$ is a simple Lie algebra, that is:\n\n1. $\\mathfrak{g}$ is not abelian, and\n2. the only ideals of $\\mathfrak{g}$ are $0$ and $\\mathfrak{g}$.\n\n## Remarks\n\nExample: $G = SU(2)$, with Lie algebra $\\mathfrak{su}(2) \\cong \\R^3$ and the cross product as bracket.\nThe bracket is not identically zero, so clause 1 holds.\nIf an ideal $I$ contains $x \\ne 0$, it contains $x \\times y$ for all $y$, and these vectors fill the plane $x^\\perp$.\nThe cross product of two independent vectors of that plane is a nonzero multiple of $x$, so $I = \\R^3$, and clause 2 holds.\n\nNon-example: $G = U(n)$ with $n \\ge 2$.\nIts Lie algebra $\\mathfrak{u}(n)$ of skew-Hermitian matrices contains the ideal $i\\R\\, I_n$ of scalar matrices, which is neither $0$ nor $\\mathfrak{u}(n)$, so clause 2 fails.\nFor $n = 1$ the group $U(1)$ is abelian, so clause 1 fails.\n\nFor connected $G$, the connected normal Lie subgroups of $G$ correspond to the ideals of $\\mathfrak{g}$.\nSo $G$ is simple exactly when it is not abelian and has no connected normal Lie subgroups other than $\\set{e}$ and $G$.\nA simple Lie group may have a nontrivial discrete centre, so it need not be a simple group in the sense of abstract group theory.\nSome authors also require the centre to be trivial or finite.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "AM7YJP8AR6",
    "kind": "definition",
    "msc": [
      "22E46",
      "22E15",
      "22E10"
    ],
    "name": "Simple Lie group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Simple_Lie_group"
      }
    ],
    "tag": "MQ9A7TZEQ0",
    "uses": [
      "7JBRGCVJRB"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/MQ9A7TZEQ0/",
  "links": [
    {
      "field": "defines",
      "target": "AM7YJP8AR6"
    },
    {
      "field": "uses",
      "target": "7JBRGCVJRB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/MQ9A7TZEQ0.md",
  "msc": [
    "22E46",
    "22E15",
    "22E10"
  ],
  "name": "Simple Lie group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MQ9/MQ9A7TZEQ0.md",
  "status": null,
  "tag": "MQ9A7TZEQ0",
  "uri": "m2://MQ9A7TZEQ0",
  "url": "https://tamnd.github.io/math2/wiki/simple-lie-group-definition/",
  "writing": {
    "tag": "MQ9A7TZEQ0",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}