{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $K$ be a field of [[characteristic|0F6ES3BRYP]] $0$.\nLet $H$ be a [[Hopf algebra|7XFYP0N84H]] over $K$ that is cocommutative and connected, which means that $K \\cdot 1$ is the only simple subcoalgebra of $H$.\nLet $P(H) = \\set{x \\in H : \\Delta(x) = x \\otimes 1 + 1 \\otimes x}$ be the space of primitive elements, a Lie algebra under the commutator $[x, y] = xy - yx$.\nThen the inclusion $P(H) \\to H$ extends to an isomorphism of Hopf algebras $U(P(H)) \\to H$, where $U(P(H))$ is the [[universal enveloping algebra|SCZ8PRVY00]] of $P(H)$.\n\n## Remarks\n\nThe map $U(P(H)) \\to H$ exists by the universal property of $U$, and it respects the coproducts because both are generated by primitive elements.\nInjectivity uses the Poincaré–Birkhoff–Witt theorem and the fact that in characteristic $0$ the primitive elements of $U(\\mathfrak{g})$ are exactly $\\mathfrak{g}$.\nSurjectivity is proved by induction along the coradical filtration of $H$.\nExample: $H = K[x]$ with $x$ primitive; then $P(H) = Kx$, an abelian Lie algebra of dimension $1$, and $U(Kx) = K[x]$.\nCharacteristic $0$ is needed: if $K$ has characteristic $p \u003e 0$, then $x^p$ is also primitive in $K[x]$, since $\\Delta(x)^p = x^p \\otimes 1 + 1 \\otimes x^p$.\nThen $U(P(K[x]))$ is a polynomial ring in the independent variables $x, x^p, x^{p^2}, \\dots$, and the map to $K[x]$ is not injective.\nConnectedness is needed: for a group $G \\ne 1$, $KG$ is cocommutative but $P(KG) = 0$, so $U(P(KG)) = K \\ne KG$.\nMilnor and Moore proved the graded version.\nA connected graded cocommutative Hopf algebra over a field of characteristic $0$ is the universal enveloping algebra of its graded Lie algebra of primitives.\n\n## Explanations\n\n### L1 human\n\nLie algebras and their universal enveloping algebras give many Hopf algebras.\nThe theorem says that, over a field of characteristic $0$, such as $\\Q$ or $\\C$, every cocommutative Hopf algebra that is connected arises in this way.\nSo such a Hopf algebra carries no information beyond the Lie algebra of its primitive elements.\n\n### L2 human\n\nThe theorem is one half of the classification of cocommutative Hopf algebras in characteristic $0$.\nOver an algebraically closed field, the Cartier–Gabriel–Kostant theorem adds group algebras: $H$ is the smash product $U(P(H)) \\# KG(H)$, where $G(H)$ is the group of group-like elements.\nIn algebraic topology, let $X$ be a simply connected space with loop space $\\Omega X$.\nThe graded version identifies the rational homology of $\\Omega X$ with the universal enveloping algebra of the Lie algebra $\\pi_*(\\Omega X) \\otimes \\Q$.\nIt is also the reason why the Hopf algebras of renormalization in physics are dual to universal enveloping algebras.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "7XFYP0N84H",
      "SCZ8PRVY00"
    ],
    "kind": "statement",
    "msc": [
      "16T05",
      "17B35",
      "57T05"
    ],
    "name": "Cartier–Milnor–Moore theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Milnor–Moore_theorem"
      }
    ],
    "tag": "DHCHJNHD02",
    "uses": [
      "7XFYP0N84H",
      "SCZ8PRVY00",
      "0F6ES3BRYP"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/DHCHJNHD02/",
  "links": [
    {
      "field": "about",
      "target": "7XFYP0N84H"
    },
    {
      "field": "about",
      "target": "SCZ8PRVY00"
    },
    {
      "field": "uses",
      "target": "7XFYP0N84H"
    },
    {
      "field": "uses",
      "target": "SCZ8PRVY00"
    },
    {
      "field": "uses",
      "target": "0F6ES3BRYP"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/DHCHJNHD02.md",
  "msc": [
    "16T05",
    "17B35",
    "57T05"
  ],
  "name": "Cartier–Milnor–Moore theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DHC/DHCHJNHD02.md",
  "status": null,
  "tag": "DHCHJNHD02",
  "uri": "m2://DHCHJNHD02",
  "url": "https://tamnd.github.io/math2/wiki/cartier-milnor-moore-theorem/",
  "writing": {
    "tag": "DHCHJNHD02",
    "kind": "statement",
    "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
      }
    ]
  }
}