{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a connected real or complex [[Lie group|7JBRGCVJRB]].\nIts [[lower central series|0K98R7A4VC]] is $C^1 G = G$ and $C^{k+1} G = [G, C^k G]$ for $k \\ge 1$, where $[G, H]$ is the subgroup generated by the commutators $g h g^{-1} h^{-1}$ with $g \\in G$ and $h \\in H$.\nThe group $G$ is a *nilpotent Lie group* if $C^{k} G = \\set{e}$ for some $k \\ge 1$, that is, if $G$ is a [[nilpotent group|0Q7MMQHC1P]].\n\n## Remarks\n\nA connected Lie group is nilpotent if and only if its Lie algebra $\\mathfrak{g}$ is a nilpotent Lie algebra, that is, the series $\\mathfrak{g} \\supseteq [\\mathfrak{g}, \\mathfrak{g}] \\supseteq [\\mathfrak{g}, [\\mathfrak{g}, \\mathfrak{g}]] \\supseteq \\cdots$ reaches $0$.\n\nExample: the group $N$ of real upper triangular $3 \\times 3$ matrices with $1$ on the diagonal.\nWriting $[a, b, c]$ for the matrix with entries $a$, $b$ above the diagonal and $c$ in the corner, a direct computation shows that the commutator of $[a, b, c]$ and $[a', b', c']$ is $[0, 0, ab' - a'b]$.\nSo $C^2 N = \\set{[0, 0, c]}$, which is the centre of $N$, and $C^3 N = \\set{e}$; $N$ is nilpotent.\n\nNon-example: the group $G$ of affine maps $x \\mapsto a x + b$ of $\\R$ with $a \u003e 0$, written $(a, b)$.\nThe commutator of $(a, 0)$ and $(1, t)$ is $(1, (a - 1)t)$, so $C^2 G$ is the group of all translations $(1, b)$, and then $C^3 G = [G, C^2 G]$ again contains every translation.\nSo $C^k G$ is the translation group for all $k \\ge 2$ and never trivial: $G$ is solvable but not nilpotent.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "DK87YN458D",
    "kind": "definition",
    "msc": [
      "22E25",
      "22E15",
      "20F18"
    ],
    "name": "Nilpotent Lie group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Nilpotent_group"
      }
    ],
    "tag": "S06VTCGCHN",
    "uses": [
      "7JBRGCVJRB",
      "0Q7MMQHC1P",
      "0K98R7A4VC"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/S06VTCGCHN/",
  "links": [
    {
      "field": "defines",
      "target": "DK87YN458D"
    },
    {
      "field": "uses",
      "target": "7JBRGCVJRB"
    },
    {
      "field": "uses",
      "target": "0Q7MMQHC1P"
    },
    {
      "field": "uses",
      "target": "0K98R7A4VC"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/S06VTCGCHN.md",
  "msc": [
    "22E25",
    "22E15",
    "20F18"
  ],
  "name": "Nilpotent Lie group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/S06/S06VTCGCHN.md",
  "status": null,
  "tag": "S06VTCGCHN",
  "uri": "m2://S06VTCGCHN",
  "url": "https://tamnd.github.io/math2/wiki/nilpotent-lie-group-definition/",
  "writing": {
    "tag": "S06VTCGCHN",
    "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
      }
    ]
  }
}