{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "279SXZFBJ2"
    },
    {
      "field": "uses",
      "target": "4R7HW6X4JY"
    },
    {
      "field": "uses",
      "target": "P6QNEK44RX"
    },
    {
      "field": "uses",
      "target": "X0X1Q5JZFK"
    }
  ],
  "body": "\n## Summary\n\nA left-invariant vector field on a [[Lie group|7JBRGCVJRB]] is a [[vector field|0299VCW1TH]] that is unchanged by every left translation.\nSuch a field is fixed by its value at the identity, so left-invariant vector fields correspond to tangent vectors at the identity.\nThey are closed under the [[Lie bracket of vector fields|68FGQEBD6J]].\nThis makes the tangent space at the identity a Lie algebra, the Lie algebra of the group.\n\n## Explanations\n\n### L1 human\n\nPick an arrow at the identity element of a Lie group.\nCarry it to every other point $g$ by multiplying on the left by $g$; you get an arrow at every point, all fitting together smoothly.\nOn the plane $\\R^2$ as an additive group, this gives the constant vector fields: the same arrow attached at every point.\nOn the circle group, it gives a field of arrows of the same length all pointing anticlockwise, or all clockwise.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22E60",
      "22E15"
    ],
    "name": "Left-invariant vector field",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lie_group"
      }
    ],
    "specializes": [
      "0299VCW1TH"
    ],
    "tag": "SNA3DVPK0C"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/SNA3DVPK0C/",
  "links": [
    {
      "field": "specializes",
      "target": "0299VCW1TH"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/SNA3DVPK0C.md",
  "msc": [
    "22E60",
    "22E15"
  ],
  "name": "Left-invariant vector field",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SNA/SNA3DVPK0C.md",
  "status": null,
  "tag": "SNA3DVPK0C",
  "uri": "m2://SNA3DVPK0C",
  "url": "https://tamnd.github.io/math2/wiki/left-invariant-vector-field/",
  "writing": {
    "tag": "SNA3DVPK0C",
    "kind": "concept",
    "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
      }
    ]
  }
}