{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ be a unital C*-algebra and let $A \\otimes A$ denote its minimal C*-tensor product with itself.\nA *compact quantum group* is a pair $(A, \\Delta)$ where $\\Delta \\colon A \\to A \\otimes A$ is a unital $*$-homomorphism, called the comultiplication, such that:\n\n1. $(\\Delta \\otimes \\id) \\circ \\Delta = (\\id \\otimes \\Delta) \\circ \\Delta$ (coassociativity);\n2. the linear spans of $\\set{\\Delta(a)(1 \\otimes b) : a, b \\in A}$ and of $\\set{\\Delta(a)(b \\otimes 1) : a, b \\in A}$ are both norm-dense in $A \\otimes A$ (cancellation).\n\n## Remarks\n\nExample: let $G$ be a [[compact|0TVJZYB6SH]] Hausdorff group and $A = C(G)$, so that $A \\otimes A = C(G \\times G)$.\nSet $\\Delta(f)(s, t) = f(st)$.\nCoassociativity is the associativity $(rs)t = r(st)$.\nThe products $\\Delta(f)(1 \\otimes g)$ are the functions $(s, t) \\mapsto f(st) g(t)$.\nSince $(s, t) \\mapsto (st, t)$ is a homeomorphism of $G \\times G$, their span is a self-adjoint subalgebra that separates points and contains $1$.\nSo it is dense by the Stone–Weierstrass theorem, and the other span is handled in the same way.\n\nNon-example: let $S = [0, 1]$ with ordinary multiplication, a compact semigroup that is not a group, and define $\\Delta$ on $C(S)$ by the same formula.\nClause 1 holds, but every function $(s, t) \\mapsto f(st) g(t)$ is constant in $s$ on the line $t = 0$.\nThis property passes to linear combinations and uniform limits, so the span is not dense, and clause 2 fails.\n\nFor commutative $A$, every compact quantum group is of the form $C(G)$ for a compact group $G$.\nEvery compact quantum group has a unique state $h$, the Haar state, with $(h \\otimes \\id)\\Delta(a) = (\\id \\otimes h)\\Delta(a) = h(a) 1$ for all $a \\in A$; it plays the role of the Haar measure.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "FHQKT6HPW2",
    "kind": "definition",
    "msc": [
      "46L67",
      "20G42",
      "58B32"
    ],
    "name": "Compact quantum group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Compact_quantum_group"
      }
    ],
    "tag": "YT35QHA4EA",
    "uses": [
      "0TVJZYB6SH"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/YT35QHA4EA/",
  "links": [
    {
      "field": "defines",
      "target": "FHQKT6HPW2"
    },
    {
      "field": "uses",
      "target": "0TVJZYB6SH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YT35QHA4EA.md",
  "msc": [
    "46L67",
    "20G42",
    "58B32"
  ],
  "name": "Compact quantum group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YT3/YT35QHA4EA.md",
  "status": null,
  "tag": "YT35QHA4EA",
  "uri": "m2://YT35QHA4EA",
  "url": "https://math2.tamnd.com/wiki/compact-quantum-group-definition/",
  "writing": {
    "tag": "YT35QHA4EA",
    "kind": "definition",
    "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
      }
    ]
  }
}