{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(M, \\omega)$ be a [[symplectic manifold|CAN490EPRT]] with a Hamiltonian action of a Lie group $G$ and a [[moment map|ZARM34FZ6Z]] $\\mu \\colon M \\to \\mathfrak{g}^*$.\nAssume that $G$ acts freely and properly on the level set $Z = \\mu^{-1}(0)$.\nThen $Z$ is a submanifold of $M$, the [[quotient|0BR3FY5Q8M]] $M /\\!\\!/ G = Z / G$ is a smooth manifold, and the quotient map $\\pi \\colon Z \\to Z / G$ is smooth.\nLet $i \\colon Z \\to M$ be the inclusion.\nThe *Marsden–Weinstein reduction* (or *symplectic quotient*) of $M$ by $G$ is the manifold $M /\\!\\!/ G$ with the unique $2$-form $\\omega_{\\mathrm{red}}$ such that\n$$\\pi^* \\omega_{\\mathrm{red}} = i^* \\omega.$$\n\n## Remarks\n\nThe Marsden–Weinstein–Meyer theorem says that $\\omega_{\\mathrm{red}}$ exists, is unique, and is symplectic, and that $\\dim M /\\!\\!/ G = \\dim M - 2 \\dim G$.\nFreeness of the action on $Z$ implies that $0$ is a regular value of $\\mu$, which is why $Z$ is a submanifold.\nProperness holds automatically when $G$ is compact.\nReduction at another value $\\lambda \\in \\mathfrak{g}^*$ uses $\\mu^{-1}(\\lambda)$ divided by the stabilizer of $\\lambda$ under the coadjoint action.\n\nExample: let $S^1$ act on $\\C^{n+1}$ by $e^{i\\theta} \\cdot z = e^{i\\theta} z$, with the standard symplectic form.\nA moment map is $\\mu(z) = -\\frac{1}{2}\\abs{z}^2 + \\frac{1}{2}$.\nThen $\\mu^{-1}(0)$ is the unit sphere $S^{2n+1}$, the circle acts freely on it, and the reduction is $\\mathbb{CP}^n$ with a multiple of the Fubini–Study form.\n\nNon-example: take the same circle action on $\\C$ with $\\mu(z) = -\\frac{1}{2}\\abs{z}^2$.\nThe level set $\\mu^{-1}(0) = \\set{0}$ is a fixed point, so the action on it is not free and the definition does not apply.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "SBJVATVH44",
    "kind": "definition",
    "msc": [
      "53D20"
    ],
    "name": "Marsden–Weinstein reduction",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Momentum_map"
      }
    ],
    "tag": "X7ZSDSXYT5",
    "uses": [
      "CAN490EPRT",
      "ZARM34FZ6Z",
      "0BR3FY5Q8M"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/X7ZSDSXYT5/",
  "links": [
    {
      "field": "defines",
      "target": "SBJVATVH44"
    },
    {
      "field": "uses",
      "target": "CAN490EPRT"
    },
    {
      "field": "uses",
      "target": "ZARM34FZ6Z"
    },
    {
      "field": "uses",
      "target": "0BR3FY5Q8M"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/X7ZSDSXYT5.md",
  "msc": [
    "53D20"
  ],
  "name": "Marsden–Weinstein reduction",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/X7Z/X7ZSDSXYT5.md",
  "status": null,
  "tag": "X7ZSDSXYT5",
  "uri": "m2://X7ZSDSXYT5",
  "url": "https://math2.tamnd.com/wiki/marsden-weinstein-reduction-definition/",
  "writing": {
    "tag": "X7ZSDSXYT5",
    "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
      }
    ]
  }
}