{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $A \\to M$ be a Lie algebroid with anchor $\\rho \\colon A \\to TM$.\nFor $x \\in M$, let $\\mathfrak{g}_x = \\ker(\\rho_x)$ be the isotropy Lie algebra at $x$, with centre $Z(\\mathfrak{g}_x)$.\nLet $\\mathcal{N}_x(A) \\subseteq Z(\\mathfrak{g}_x)$ be the monodromy group at $x$.\nIt is the set of $v \\in Z(\\mathfrak{g}_x)$ such that the constant $A$-path with value $v$ is $A$-homotopic to the zero $A$-path at $x$.\nFix a continuous [[norm|08NYZ2FS6B]] on the fibres of $A$, and let $r(x)$ be the distance from $0$ to $\\mathcal{N}_x(A) \\setminus \\set{0}$, with $r(x) = \\infty$ when $\\mathcal{N}_x(A) = \\set{0}$.\nThen $A$ is isomorphic to the Lie algebroid of some [[Lie groupoid|HNVW0E732J]] if and only if, for every $x \\in M$,\n\n1. $\\mathcal{N}_x(A)$ is a discrete subset of $A_x$, and\n2. $\\liminf_{y \\to x} r(y) \u003e 0$.\n\n## Remarks\n\nCrainic and Fernandes build a candidate groupoid $\\mathcal{G}(A)$, the Weinstein groupoid, as the quotient of the space of $A$-paths by $A$-homotopy.\nIt is always a topological groupoid, and $A$ is integrable exactly when $\\mathcal{G}(A)$ is smooth; it is then the unique integration whose source fibres are simply connected.\nThe monodromy groups measure the failure of smoothness, and the two conditions say that they stay uniformly discrete as the point moves.\nThe condition does not depend on the chosen norm.\n\nExample: when $M$ is a point, $A$ is a finite-dimensional Lie algebra and the monodromy groups are trivial.\nSo every Lie algebra integrates to a Lie group, which is Lie's third theorem.\nUnlike Lie algebras, Lie algebroids need not be integrable, and the theorem identifies the obstruction.\n\n## Explanations\n\n### L1 human\n\nEvery Lie group has a Lie algebra, and every finite-dimensional Lie algebra comes from a Lie group.\nLie algebroids play the role of Lie algebras for Lie groupoids, but the second half of this correspondence can fail for them.\nThe theorem gives an exact test for when a Lie algebroid comes from a Lie groupoid.\nThe test checks that certain discrete groups attached to each point do not shrink towards zero as the point moves.\n\n### L2 human\n\nThe theorem settles the integrability problem for Lie algebroids, which asks for the analogue of Lie's third theorem.\nIt applies to Poisson manifolds through their cotangent Lie algebroids, where integrability means the existence of a symplectic groupoid.\nThe monodromy group at $x$ is the image of a homomorphism defined on the second homotopy group of the orbit through $x$.\nSo integrability is automatic when every orbit has trivial second homotopy group.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "HNVW0E732J"
    ],
    "kind": "statement",
    "msc": [
      "58H05",
      "22A22",
      "53D17"
    ],
    "name": "Crainic–Fernandes integrability theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lie_algebroid"
      }
    ],
    "tag": "5ZNN348M6X",
    "uses": [
      "HNVW0E732J",
      "08NYZ2FS6B"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/5ZNN348M6X/",
  "links": [
    {
      "field": "about",
      "target": "HNVW0E732J"
    },
    {
      "field": "uses",
      "target": "HNVW0E732J"
    },
    {
      "field": "uses",
      "target": "08NYZ2FS6B"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5ZNN348M6X.md",
  "msc": [
    "58H05",
    "22A22",
    "53D17"
  ],
  "name": "Crainic–Fernandes integrability theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5ZN/5ZNN348M6X.md",
  "status": null,
  "tag": "5ZNN348M6X",
  "uri": "m2://5ZNN348M6X",
  "url": "https://tamnd.github.io/math2/wiki/crainic-fernandes-integrability-theorem/",
  "writing": {
    "tag": "5ZNN348M6X",
    "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
      }
    ]
  }
}