{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1NX95TTMFT"
    },
    {
      "field": "uses",
      "target": "263GGJFA21"
    },
    {
      "field": "uses",
      "target": "279SXZFBJ2"
    },
    {
      "field": "uses",
      "target": "2B3BWV1HPC"
    },
    {
      "field": "uses",
      "target": "2K8BZRGE8R"
    },
    {
      "field": "uses",
      "target": "3MXR03096V"
    },
    {
      "field": "uses",
      "target": "3NJGY8QSTE"
    },
    {
      "field": "uses",
      "target": "4R7HW6X4JY"
    },
    {
      "field": "uses",
      "target": "4RXXE3BB6W"
    },
    {
      "field": "uses",
      "target": "5GDFYDF2R2"
    },
    {
      "field": "uses",
      "target": "5X6G3Q5CSQ"
    },
    {
      "field": "uses",
      "target": "707DEWCCFX"
    },
    {
      "field": "uses",
      "target": "8MEDVAEERQ"
    },
    {
      "field": "uses",
      "target": "8XAT77QQTN"
    },
    {
      "field": "uses",
      "target": "A399EBX2YD"
    },
    {
      "field": "uses",
      "target": "B6K478R9XE"
    },
    {
      "field": "uses",
      "target": "CZCWFPNS4N"
    },
    {
      "field": "uses",
      "target": "D0Q6GE7FG4"
    },
    {
      "field": "uses",
      "target": "DDQJGJGQGC"
    },
    {
      "field": "uses",
      "target": "DS2AS9TSZP"
    },
    {
      "field": "uses",
      "target": "DWMQ3V205V"
    },
    {
      "field": "uses",
      "target": "DYXEBVDC6W"
    },
    {
      "field": "uses",
      "target": "E0ZMRTA42A"
    },
    {
      "field": "uses",
      "target": "F4ZYFEKKZW"
    },
    {
      "field": "uses",
      "target": "GC7T4VW3N1"
    },
    {
      "field": "uses",
      "target": "H51S5V6XYJ"
    },
    {
      "field": "uses",
      "target": "JC9R1073M9"
    },
    {
      "field": "uses",
      "target": "JTXSC68JZE"
    },
    {
      "field": "uses",
      "target": "KCP52TPREW"
    },
    {
      "field": "uses",
      "target": "M13HAQSHEM"
    },
    {
      "field": "uses",
      "target": "ME4ZSRAT5Z"
    },
    {
      "field": "uses",
      "target": "MQ9A7TZEQ0"
    },
    {
      "field": "uses",
      "target": "NR70GHZ37D"
    },
    {
      "field": "uses",
      "target": "P43WKB4372"
    },
    {
      "field": "uses",
      "target": "P52TCSHP7R"
    },
    {
      "field": "uses",
      "target": "P6QNEK44RX"
    },
    {
      "field": "uses",
      "target": "QEHV2DAHSW"
    },
    {
      "field": "uses",
      "target": "S06VTCGCHN"
    },
    {
      "field": "uses",
      "target": "S5VTE2R86M"
    },
    {
      "field": "uses",
      "target": "T866256Y1W"
    },
    {
      "field": "uses",
      "target": "V0SGQ7E092"
    },
    {
      "field": "uses",
      "target": "V3XM2N7K0X"
    },
    {
      "field": "uses",
      "target": "WTCNKST134"
    },
    {
      "field": "uses",
      "target": "X0X1Q5JZFK"
    },
    {
      "field": "uses",
      "target": "X898DPBSY7"
    },
    {
      "field": "uses",
      "target": "XJF5V9WHKZ"
    },
    {
      "field": "uses",
      "target": "YHGS413E8B"
    },
    {
      "field": "uses",
      "target": "Z367AGJRWH"
    },
    {
      "field": "uses",
      "target": "ZZYCV5EX9J"
    },
    {
      "field": "specializes",
      "target": "36KSZ7GA5S"
    },
    {
      "field": "specializes",
      "target": "42ZCE14AD2"
    },
    {
      "field": "specializes",
      "target": "9CNSVFQ801"
    },
    {
      "field": "specializes",
      "target": "9TFC6ADC1C"
    },
    {
      "field": "illustrates",
      "target": "720FQEBT68"
    },
    {
      "field": "instance_of",
      "target": "8B0JM2YDE2"
    },
    {
      "field": "instance_of",
      "target": "8ZQ9TA4Q2Q"
    },
    {
      "field": "instance_of",
      "target": "J58ED1GV6N"
    },
    {
      "field": "instance_of",
      "target": "M2900DMEW6"
    },
    {
      "field": "instance_of",
      "target": "Q0GBCB9RA4"
    },
    {
      "field": "instance_of",
      "target": "QWK3BYVJ9W"
    },
    {
      "field": "about",
      "target": "DDQJGJGQGC"
    },
    {
      "field": "about",
      "target": "DYXEBVDC6W"
    },
    {
      "field": "about",
      "target": "M13HAQSHEM"
    },
    {
      "field": "about",
      "target": "NR70GHZ37D"
    },
    {
      "field": "about",
      "target": "P6QNEK44RX"
    },
    {
      "field": "about",
      "target": "XJF5V9WHKZ"
    },
    {
      "field": "about",
      "target": "YHGS413E8B"
    },
    {
      "field": "defines",
      "target": "K14ZBKYT81"
    }
  ],
  "body": "\n## Summary\n\nA Lie group is a group that is also a smooth manifold, in such a way that multiplication and inversion are smooth maps.\nIt is the natural setting for continuous symmetry: rotations, translations, matrix groups and the symmetry groups of physics are all Lie groups.\nIts structure near the identity is captured by a finite-dimensional Lie algebra, which turns many questions about the group into linear algebra.\n\n## Explanations\n\n### L1 human\n\nA Lie group is a family of symmetries that can be varied smoothly, like the rotations of a plane through any angle.\nEach element is a point of a smooth space, and you can move from one element to a nearby one by a small change.\nThe circle of complex numbers of absolute value $1$, with multiplication, is a simple example: it is a smooth curve, and multiplying two points of it just adds their angles.\nThe invertible $n \\times n$ real matrices form another example, a group that is an open subset of the space of all matrices.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22E15",
      "22E10"
    ],
    "name": "Lie group",
    "same_as": {
      "wikidata": "Q622679",
      "wikipedia": "Lie group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lie_group"
      }
    ],
    "specializes": [
      "92700QAB5Y"
    ],
    "tag": "7JBRGCVJRB"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/7JBRGCVJRB/",
  "links": [
    {
      "field": "specializes",
      "target": "92700QAB5Y"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7JBRGCVJRB.md",
  "msc": [
    "22E15",
    "22E10"
  ],
  "name": "Lie group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7JB/7JBRGCVJRB.md",
  "status": null,
  "tag": "7JBRGCVJRB",
  "uri": "m2://7JBRGCVJRB",
  "url": "https://tamnd.github.io/math2/wiki/lie-group/",
  "writing": {
    "tag": "7JBRGCVJRB",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}