{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2EPRMNK1G2"
    },
    {
      "field": "uses",
      "target": "720FQEBT68"
    },
    {
      "field": "uses",
      "target": "E5P8AG6CK7"
    },
    {
      "field": "uses",
      "target": "FJZXB1N7TP"
    },
    {
      "field": "uses",
      "target": "YX5FFSCR8P"
    }
  ],
  "body": "\n## Example\n\nLet $n \\ge 1$.\nThe *special unitary group* is\n$$SU(n) = \\set{U \\in M_n(\\C) : U^* U = I_n, \\ \\det U = 1},$$\nwhere $U^*$ is the conjugate transpose of $U$ and $I_n$ is the identity matrix.\nIt is the kernel of the [[determinant|00KDZRW4C0]] $\\det \\colon U(n) \\to U(1)$ on the [[unitary group|0R9R0Y1HDK]], so it is a normal subgroup of $U(n)$.\n\nCheck that it is a [[compact Lie group|9TFC6ADC1C]].\nThe two conditions are polynomial equations in the real and imaginary parts of the entries, so $SU(n)$ is closed in $GL_n(\\C)$.\nBy the closed subgroup theorem it is a [[Lie group|7JBRGCVJRB]].\nEach column of a unitary matrix is a unit vector, so $SU(n)$ is bounded in $M_n(\\C)$, hence compact.\n\n## Remarks\n\nIts Lie algebra $\\mathfrak{su}(n)$ consists of the skew-Hermitian $n \\times n$ matrices with trace $0$, so its real dimension is $n^2 - 1$.\n$SU(n)$ is connected and simply connected.\nIts centre is the set of scalar matrices $\\omega I_n$ with $\\omega^n = 1$, a cyclic group of order $n$.\nFor $n \\ge 2$ its Lie algebra is simple, of type $A_{n-1}$.\nIn Mathlib this group is `Matrix.specialUnitaryGroup`.\n\n## Explanations\n\n### L1 human\n\nThe special unitary group collects the complex matrices that keep lengths and also have determinant $1$.\nThe unitary group is a little larger: its determinants are all complex numbers of absolute value $1$.\nFixing the determinant to $1$ removes one real degree of freedom, a single overall phase.\nFor $n = 1$ the group is trivial, and for $n = 2$ it is a copy of the $3$-sphere.\nThe groups $SU(2)$ and $SU(3)$ are central in particle physics.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "9TFC6ADC1C"
    ],
    "kind": "example",
    "msc": [
      "22E15",
      "22C05",
      "20G20"
    ],
    "name": "Special unitary group SU(n)",
    "same_as": {
      "wikidata": "Q684363",
      "wikipedia": "Special unitary group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Special_unitary_group"
      }
    ],
    "tag": "CZCWFPNS4N",
    "uses": [
      "0R9R0Y1HDK",
      "00KDZRW4C0",
      "7JBRGCVJRB"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/CZCWFPNS4N/",
  "links": [
    {
      "field": "instance_of",
      "target": "9TFC6ADC1C"
    },
    {
      "field": "uses",
      "target": "0R9R0Y1HDK"
    },
    {
      "field": "uses",
      "target": "00KDZRW4C0"
    },
    {
      "field": "uses",
      "target": "7JBRGCVJRB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CZCWFPNS4N.md",
  "msc": [
    "22E15",
    "22C05",
    "20G20"
  ],
  "name": "Special unitary group SU(n)",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CZC/CZCWFPNS4N.md",
  "status": null,
  "tag": "CZCWFPNS4N",
  "uri": "m2://CZCWFPNS4N",
  "url": "https://tamnd.github.io/math2/wiki/special-unitary-group-su-n/",
  "writing": {
    "tag": "CZCWFPNS4N",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}