{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "YT35QHA4EA"
    }
  ],
  "body": "\n## Summary\n\nA compact quantum group is a unital C*-algebra with a coassociative comultiplication that satisfies a cancellation condition.\nIt was introduced by Stanisław Woronowicz as the noncommutative analogue of a compact group, seen through its algebra of continuous functions.\nEvery compact group gives one, and there are genuinely quantum examples such as $SU_q(2)$.\nMuch of the theory of compact groups survives: there is a unique Haar state, and finite-dimensional representations behave as in Peter–Weyl theory.\n\n## Explanations\n\n### L1 human\n\nA compact group $G$ can be described by the algebra $C(G)$ of continuous functions on it.\nThe group law becomes a map $\\Delta$ that sends a function $f$ to the function $(s, t) \\mapsto f(st)$ of two variables.\nA compact quantum group keeps this kind of map but allows the algebra to be noncommutative.\nFor example, the circle group gives the algebra $C(S^1)$ with $\\Delta(z) = z \\otimes z$ for the coordinate function $z$.\nDeforming such examples with a parameter $q$ gives quantum groups that have no underlying set of points.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46L67",
      "20G42",
      "58B32"
    ],
    "name": "Compact quantum group",
    "same_as": {
      "wikidata": "Q5155316",
      "wikipedia": "Compact quantum group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Compact_quantum_group"
      }
    ],
    "tag": "FHQKT6HPW2"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/FHQKT6HPW2/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/FHQKT6HPW2.md",
  "msc": [
    "46L67",
    "20G42",
    "58B32"
  ],
  "name": "Compact quantum group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FHQ/FHQKT6HPW2.md",
  "status": null,
  "tag": "FHQKT6HPW2",
  "uri": "m2://FHQKT6HPW2",
  "url": "https://math2.tamnd.com/wiki/compact-quantum-group/",
  "writing": {
    "tag": "FHQKT6HPW2",
    "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
      }
    ]
  }
}