{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "9CMAJWWBNZ"
    },
    {
      "field": "uses",
      "target": "ESRDNKF6WJ"
    }
  ],
  "body": "\n## Summary\n\nThe skew group ring $R * G$ of a group $G$ acting on a [[ring|0B1261B5ED]] $R$ by automorphisms is like the [[group ring|4NMTSX0XDS]] $R[G]$.\nThe difference is that group elements and coefficients do not commute: $g r = g(r) g$.\nIt is the algebraic counterpart of the C*-algebra [[crossed product|8HG8TS46A6]].\nIt turns a [[group action|0WCY42EV9V]] on a ring into a single ring, and its modules are the $R$-modules with a compatible action of $G$.\nIt is used in invariant theory, Galois theory of rings and the representation theory of finite groups.\n\n## Explanations\n\n### L1 human\n\nStart with a ring $R$ and a group $G$ that moves the elements of $R$ around by symmetries.\nForm sums $\\sum r_g g$ as in the group ring.\nWhen a group element passes a coefficient from left to right, it acts on it: $g r = g(r) g$.\nIf every $g \\in G$ acts as the identity map of $R$, then $g r = r g$ for all $g$ and $r$, and this is the group ring.\nFor the complex numbers with complex conjugation, the skew group ring is the ring of $2 \\times 2$ real matrices.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "16S35",
      "16W22"
    ],
    "name": "Skew group ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Crossed_product"
      }
    ],
    "specializes": [
      "D1SA8CHJV0"
    ],
    "tag": "PCKK4SFZEK"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/PCKK4SFZEK/",
  "links": [
    {
      "field": "specializes",
      "target": "D1SA8CHJV0"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/PCKK4SFZEK.md",
  "msc": [
    "16S35",
    "16W22"
  ],
  "name": "Skew group ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PCK/PCKK4SFZEK.md",
  "status": null,
  "tag": "PCKK4SFZEK",
  "uri": "m2://PCKK4SFZEK",
  "url": "https://tamnd.github.io/math2/wiki/skew-group-ring/",
  "writing": {
    "tag": "PCKK4SFZEK",
    "kind": "concept",
    "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
      }
    ]
  }
}