{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] with [[modular function|63TX60RTRQ]] $\\Delta \\colon G \\to (0, \\infty)$.\nThe group $G$ is *unimodular* if $\\Delta(g) = 1$ for every $g \\in G$.\nEquivalently, every left [[Haar measure|VY0HXD3GC4]] $\\mu$ on $G$ is also right-invariant: $\\mu(Ag) = \\mu(A)$ for every $g \\in G$ and every Borel set $A$.\n\n## Remarks\n\nExample: every compact group $K$ is unimodular.\nThe modular function is a continuous homomorphism, so $\\Delta(K)$ is a compact subgroup of $(0, \\infty)$.\nThe only compact subgroup of $(0, \\infty)$ is $\\set{1}$, so $\\Delta \\equiv 1$.\n\nNon-example: the group of maps $x \\mapsto ax + b$ with $a \u003e 0$ and $b \\in \\R$ has modular function $\\Delta(a, b) = 1/a$, which is not identically $1$.\n\nEvery abelian group is unimodular, since $Ag = gA$ for all $g$.\nEvery discrete group is unimodular, since counting measure is invariant on both sides.\nOn a unimodular group, $\\int_G f(x^{-1}) \\, d\\mu(x) = \\int_G f(x) \\, d\\mu(x)$ for every integrable $f$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "CXG6JZGMCK",
    "kind": "definition",
    "msc": [
      "22D05",
      "28C10"
    ],
    "name": "Unimodular group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Haar_measure"
      }
    ],
    "tag": "XPCTJTBSQ0",
    "uses": [
      "AHV05ZBCYW",
      "VY0HXD3GC4",
      "63TX60RTRQ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/XPCTJTBSQ0/",
  "links": [
    {
      "field": "defines",
      "target": "CXG6JZGMCK"
    },
    {
      "field": "uses",
      "target": "AHV05ZBCYW"
    },
    {
      "field": "uses",
      "target": "VY0HXD3GC4"
    },
    {
      "field": "uses",
      "target": "63TX60RTRQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/XPCTJTBSQ0.md",
  "msc": [
    "22D05",
    "28C10"
  ],
  "name": "Unimodular group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XPC/XPCTJTBSQ0.md",
  "status": null,
  "tag": "XPCTJTBSQ0",
  "uri": "m2://XPCTJTBSQ0",
  "url": "https://tamnd.github.io/math2/wiki/unimodular-group-definition/",
  "writing": {
    "tag": "XPCTJTBSQ0",
    "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
      }
    ]
  }
}