{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2EPRMNK1G2"
    },
    {
      "field": "uses",
      "target": "MKT1263M0A"
    }
  ],
  "body": "\n## Example\n\nLet $\\mathbb{O}$ be the [[octonions|1HRMT3DSX2]], the $8$-dimensional real normed division algebra.\nThe compact Lie group $G_2$ is the automorphism group of $\\mathbb{O}$:\n$$G_2 = \\set{g \\in GL(\\mathbb{O}) : g(xy) = g(x)\\, g(y) \\text{ for all } x, y \\in \\mathbb{O}}.$$\nIt is a closed subgroup of $GL_8(\\R)$, since it is cut out by polynomial equations, so it is a Lie group.\nEach $g \\in G_2$ fixes $1$ and preserves the octonion norm, so it maps the $7$-dimensional space $\\operatorname{Im} \\mathbb{O}$ of imaginary octonions to itself by a rotation.\nThis makes $G_2$ a closed subgroup of [[SO(7)|0CGYNDCWRR]], hence compact.\n\n$G_2$ is a connected and simply connected [[simple Lie group|AM7YJP8AR6]] of dimension $14$ and rank $2$, with trivial centre.\n\n## Remarks\n\n$G_2$ is the smallest of the five exceptional compact simple Lie groups $G_2, F_4, E_6, E_7, E_8$.\nIts Dynkin diagram has two nodes joined by a triple edge, and its root system has $12$ roots, of two lengths whose squares have ratio $3$.\nIt acts transitively on the unit sphere $S^6$ of $\\operatorname{Im} \\mathbb{O}$, with stabilizer isomorphic to [[SU(3)|CZCWFPNS4N]], so $G_2 / SU(3) \\cong S^6$.\nThe name $G_2$ also denotes the complex simple Lie group of the same type and its noncompact split real form.\nRiemannian $7$-manifolds whose holonomy group lies in $G_2$ are called $G_2$-manifolds.\n\n## Explanations\n\n### L1 human\n\nThe classical Lie groups come in infinite families of matrix groups, while $G_2$ is one of five exceptional groups outside those families.\nIt is the symmetry group of the octonions, the largest of the four normed division algebras.\nIt has dimension $14$, so a symmetry of the octonions is fixed by $14$ real parameters.\nEvery such symmetry is a rotation of the $7$ imaginary directions, but most rotations of those directions do not respect octonion multiplication.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "AM7YJP8AR6"
    ],
    "kind": "example",
    "msc": [
      "22E15",
      "17B25",
      "22C05"
    ],
    "name": "Exceptional Lie group G2",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/G2_(mathematics)"
      }
    ],
    "tag": "YX5FFSCR8P",
    "uses": [
      "1HRMT3DSX2",
      "0CGYNDCWRR",
      "CZCWFPNS4N"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/YX5FFSCR8P/",
  "links": [
    {
      "field": "instance_of",
      "target": "AM7YJP8AR6"
    },
    {
      "field": "uses",
      "target": "1HRMT3DSX2"
    },
    {
      "field": "uses",
      "target": "0CGYNDCWRR"
    },
    {
      "field": "uses",
      "target": "CZCWFPNS4N"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YX5FFSCR8P.md",
  "msc": [
    "22E15",
    "17B25",
    "22C05"
  ],
  "name": "Exceptional Lie group G2",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YX5/YX5FFSCR8P.md",
  "status": null,
  "tag": "YX5FFSCR8P",
  "uri": "m2://YX5FFSCR8P",
  "url": "https://tamnd.github.io/math2/wiki/exceptional-lie-group-g2/",
  "writing": {
    "tag": "YX5FFSCR8P",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}