{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "5X6G3Q5CSQ"
    }
  ],
  "body": "\n## Summary\n\nA Lie supergroup is a group object in the category of [[supermanifolds|Z0SB9HQBAJ]]: a supermanifold with multiplication, inverse and unit morphisms satisfying the group laws.\nIt is the analogue of a [[Lie group|7JBRGCVJRB]] in which some coordinates are anticommuting, or odd.\nIts infinitesimal version is a Lie superalgebra, and Lie supergroups describe supersymmetry in physics.\n\n## Explanations\n\n### L1 human\n\nIn a supermanifold, besides ordinary coordinates there are odd coordinates $\\theta$ that anticommute, so $\\theta_1\\theta_2 = -\\theta_2\\theta_1$ and $\\theta^2 = 0$.\nA Lie supergroup is such a space with a smooth group law.\nA simple example has one ordinary coordinate $t$ and one odd coordinate $\\theta$, with product $(t, \\theta)(t', \\theta') = (t + t' + \\theta\\theta', \\theta + \\theta')$.\nForgetting the odd coordinates leaves an ordinary Lie group, here the real line under addition.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "58A50",
      "22E99"
    ],
    "name": "Lie supergroup",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Supergroup_(physics)"
      }
    ],
    "tag": "9DQB442K3Y"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/9DQB442K3Y/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/9DQB442K3Y.md",
  "msc": [
    "58A50",
    "22E99"
  ],
  "name": "Lie supergroup",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9DQ/9DQB442K3Y.md",
  "status": null,
  "tag": "9DQB442K3Y",
  "uri": "m2://9DQB442K3Y",
  "url": "https://tamnd.github.io/math2/wiki/lie-supergroup/",
  "writing": {
    "tag": "9DQB442K3Y",
    "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
      }
    ]
  }
}