{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a Lie group with Lie algebra $\\mathfrak{g}$, and let $\\mathbb{K}$ be $\\R$ or $\\C$.\nFor $k \\ge 0$, let $I^k(G)$ be the space of symmetric $k$-linear maps $f : \\mathfrak{g}^k \\to \\mathbb{K}$ that are invariant under the adjoint action: $f(\\mathrm{Ad}_g X_1, \\dots, \\mathrm{Ad}_g X_k) = f(X_1, \\dots, X_k)$ for all $g \\in G$.\nLet $\\pi : P \\to M$ be a principal $G$-bundle with a [[connection form|NJ21PN5ZGJ]] $\\omega$ and [[curvature form|ZAJFW1S078]] $\\Omega$.\nFor $f \\in I^k(G)$, the $2k$-form $f(\\Omega, \\dots, \\Omega)$ on $P$ (formed with the wedge product) equals $\\pi^* \\alpha_f$ for a unique closed $2k$-form $\\alpha_f$ on $M$.\nThe *Chern–Weil homomorphism* is the map\n$$I^*(G) = \\bigoplus_{k \\ge 0} I^k(G) \\to H^{\\mathrm{even}}_{\\mathrm{dR}}(M; \\mathbb{K}), \\quad f \\mapsto [\\alpha_f],$$\ninto [[de Rham cohomology|GX74C07T0G]], which does not depend on the choice of $\\omega$.\n\n## Remarks\n\nThe form $f(\\Omega, \\dots, \\Omega)$ descends to $M$ because $\\Omega$ vanishes on vertical vectors and $f$ is $\\mathrm{Ad}$-invariant; it is closed by the Bianchi identity $d\\Omega = [\\Omega \\wedge \\omega]$ and invariance.\nFor two connections $\\omega_0, \\omega_1$, the forms $\\alpha_f$ differ by the exterior derivative of a transgression form built from the path $\\omega_t = (1-t)\\omega_0 + t\\omega_1$; so the class is independent of $\\omega$.\nWith the product of symmetric multilinear forms, $I^*(G)$ is a commutative algebra and the map is an algebra homomorphism.\nExample: for $G = \\mathrm{U}(1)$, $\\mathfrak{g} = i\\R$ and $f(X) = \\frac{i}{2\\pi} X$, the class $[\\alpha_f]$ is the first Chern class of the associated line bundle.\nFor $G = \\mathrm{U}(n)$, $\\det\\big(I + \\frac{i}{2\\pi} \\Omega\\big)$ gives the total Chern class, and its images are the [[characteristic classes|7HACJC0JE2]] with real coefficients.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "82ZPRD39J7",
    "kind": "definition",
    "msc": [
      "53C05",
      "57R20"
    ],
    "name": "Chern–Weil homomorphism",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Chern%E2%80%93Weil_homomorphism"
      }
    ],
    "tag": "FBFFS10WGS",
    "uses": [
      "NJ21PN5ZGJ",
      "ZAJFW1S078",
      "GX74C07T0G",
      "7HACJC0JE2"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/FBFFS10WGS/",
  "links": [
    {
      "field": "defines",
      "target": "82ZPRD39J7"
    },
    {
      "field": "uses",
      "target": "NJ21PN5ZGJ"
    },
    {
      "field": "uses",
      "target": "ZAJFW1S078"
    },
    {
      "field": "uses",
      "target": "GX74C07T0G"
    },
    {
      "field": "uses",
      "target": "7HACJC0JE2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/FBFFS10WGS.md",
  "msc": [
    "53C05",
    "57R20"
  ],
  "name": "Chern–Weil homomorphism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FBF/FBFFS10WGS.md",
  "status": null,
  "tag": "FBFFS10WGS",
  "uri": "m2://FBFFS10WGS",
  "url": "https://math2.tamnd.com/wiki/chern-weil-homomorphism-definition/",
  "writing": {
    "tag": "FBFFS10WGS",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}