{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "8XAT77QQTN"
    },
    {
      "field": "about",
      "target": "DRY1HXP1W9"
    },
    {
      "field": "uses",
      "target": "DRY1HXP1W9"
    }
  ],
  "body": "\n## Summary\n\nThe cohomology of a [[Lie group|7JBRGCVJRB]] $G$ is the [[singular cohomology|6N7BC70CX0]] of its underlying manifold, with the extra structure given by the group multiplication.\nThe multiplication makes it a Hopf algebra, which strongly restricts its form.\nFor a compact connected group, real cohomology can be computed from invariant differential forms, that is, from the cohomology of the Lie algebra.\nThis turns a topological question into a question of linear algebra.\n\n## Explanations\n\n### L1 human\n\nCohomology counts the holes of a space in each dimension.\nA Lie group is a space, so one can ask about its holes.\nThe circle has one hole of dimension $1$, and the group $\\mathrm{SU}(2)$, a $3$-sphere, has one hole of dimension $3$.\nBecause a Lie group looks the same at every point, its holes can be detected by forms that are invariant under the group.\nThose forms are determined by their values at the identity, where they become algebra on the Lie algebra.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "57T10",
      "17B56"
    ],
    "name": "Cohomology of Lie groups",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lie_algebra_cohomology"
      }
    ],
    "tag": "HKZABKRPN6"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/HKZABKRPN6/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/HKZABKRPN6.md",
  "msc": [
    "57T10",
    "17B56"
  ],
  "name": "Cohomology of Lie groups",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HKZ/HKZABKRPN6.md",
  "status": null,
  "tag": "HKZABKRPN6",
  "uri": "m2://HKZABKRPN6",
  "url": "https://tamnd.github.io/math2/wiki/cohomology-of-lie-groups/",
  "writing": {
    "tag": "HKZABKRPN6",
    "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
      }
    ]
  }
}