{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Description\n\nLet $F$ be a field and let $n \\ge 1$.\nThe *special linear group* is the set\n$$SL_n(F) = \\{A \\in GL_n(F) : \\det A = 1\\}$$\nof $n \\times n$ matrices over $F$ with [[determinant|00KDZRW4C0]] $1$, with matrix multiplication.\nIt is a subgroup of the [[general linear group|0FCHE902D7]] $GL_n(F)$.\n\n## Remarks\n\nIt is the kernel of the group homomorphism $\\det \\colon GL_n(F) \\to F^\\times$, so it is a normal subgroup.\nThe quotient $GL_n(F) / SL_n(F)$ is isomorphic to $F^\\times$.\nOver $\\R$, its elements are the linear maps that keep volume and orientation.\n$SL_2(\\Z)$ is the analogous group of integer matrices with determinant $1$.\n\n## Explanations\n\n### L1 human\n\nThe special linear group collects the matrices with determinant $1$.\nIn the real case, these maps keep the volume of every region, and they do not flip orientation.\nShears, such as $\\begin{pmatrix} 1 \u0026 t \\\\ 0 \u0026 1 \\end{pmatrix}$, are typical elements.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "instance_of": [
      "G5S1M9"
    ],
    "kind": "object",
    "msc": [
      "15A15",
      "20G15"
    ],
    "name": "Special linear group",
    "same_as": {
      "wikidata": "Q2140900",
      "wikipedia": "Special linear group"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Special_linear_group"
      }
    ],
    "tag": "0V9PMN0RBH",
    "uses": [
      "0C3CFHDN6R",
      "0FCHE902D7"
    ]
  },
  "kind": "object",
  "link": "https://math2.tamnd.com/node/0V9PMN0RBH/",
  "links": [
    {
      "field": "instance_of",
      "target": "G5S1M9"
    },
    {
      "field": "uses",
      "target": "0C3CFHDN6R"
    },
    {
      "field": "uses",
      "target": "0FCHE902D7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0V9PMN0RBH.md",
  "msc": [
    "15A15",
    "20G15"
  ],
  "name": "Special linear group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0V9/0V9PMN0RBH.md",
  "status": null,
  "tag": "0V9PMN0RBH",
  "uri": "m2://0V9PMN0RBH",
  "url": "https://math2.tamnd.com/wiki/special-linear-group/",
  "writing": {
    "tag": "0V9PMN0RBH",
    "kind": "object",
    "level": "W1",
    "next": [
      {
        "what": "section Invariants",
        "note": "a table of the main invariants",
        "have": 0,
        "want": 1
      }
    ]
  }
}