{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "BM3TWJ0D4P"
    },
    {
      "field": "uses",
      "target": "M9DY1E7RKS"
    },
    {
      "field": "uses",
      "target": "PG1CVVN7KA"
    }
  ],
  "body": "\n## Statement\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center and Lie algebra $\\mathfrak{g}$.\nLet $\\mathfrak{g} = \\mathfrak{k} \\oplus \\mathfrak{p}$ be the Cartan decomposition for a Cartan involution, as in the [[Cartan decomposition|A5JB9JQKR0]].\nLet $K$ be the connected subgroup with Lie algebra $\\mathfrak{k}$.\nLet $\\mathfrak{a} \\subseteq \\mathfrak{p}$ be a maximal abelian subspace.\nFor a linear form $\\lambda \\ne 0$ on $\\mathfrak{a}$, put $\\mathfrak{g}_\\lambda = \\set{X \\in \\mathfrak{g} : [H, X] = \\lambda(H) X \\text{ for all } H \\in \\mathfrak{a}}$, and let $\\Sigma$ be the set of $\\lambda$ with $\\mathfrak{g}_\\lambda \\ne 0$, the restricted roots.\nFix $H_0 \\in \\mathfrak{a}$ with $\\lambda(H_0) \\ne 0$ for all $\\lambda \\in \\Sigma$, put $\\Sigma^+ = \\set{\\lambda \\in \\Sigma : \\lambda(H_0) \u003e 0}$ and $\\mathfrak{n} = \\bigoplus_{\\lambda \\in \\Sigma^+} \\mathfrak{g}_\\lambda$.\nLet $A$ and $N$ be the connected subgroups of $G$ with Lie algebras $\\mathfrak{a}$ and $\\mathfrak{n}$.\nThen $A$ is abelian, $N$ is [[nilpotent|DK87YN458D]], both are closed and simply connected, and the map\n$$K \\times A \\times N \\to G, \\quad (k, a, n) \\mapsto kan,$$\nis a diffeomorphism.\n\n## Remarks\n\nOn the Lie algebra level, $\\mathfrak{g} = \\mathfrak{k} \\oplus \\mathfrak{a} \\oplus \\mathfrak{n}$ as vector spaces; this follows from the restricted root space decomposition and the fact that $\\theta$ maps $\\mathfrak{g}_\\lambda$ to $\\mathfrak{g}_{-\\lambda}$.\nOne then shows that $AN$ is a closed simply connected solvable subgroup, and that $K \\times AN \\to G$ is a bijective local diffeomorphism.\n\nExample: $G = \\mathrm{SL}_n(\\R)$ with $K = \\mathrm{SO}(n)$.\nHere $A$ is the group of diagonal matrices with positive entries and determinant $1$, and $N$ is the group of upper triangular matrices with $1$ on the diagonal.\nThe decomposition $g = kan$ is the Gram–Schmidt process applied to the columns of $g$, also known as the QR decomposition.\n\n## Explanations\n\n### L1 human\n\nEvery element of a semisimple Lie group is, in exactly one way, a rotation times a scaling times a shear.\nFor matrices, the scaling is a positive diagonal matrix and the shear is an upper triangular matrix with ones on the diagonal.\nThis is what Gram–Schmidt orthogonalisation does to the columns of an invertible matrix.\n\n### L2 human\n\nThe Iwasawa decomposition shows that $G/K$ is diffeomorphic to the solvable group $AN$.\nSo the symmetric space $G/K$ is a solvable Lie group with a left-invariant metric.\nThe subgroup $MAN$, with $M$ the centralizer of $\\mathfrak{a}$ in $K$, is the minimal parabolic subgroup.\nInducing to $G$ the representations of $MAN$ that are trivial on $N$ gives the principal series representations.\nThe dimension of $\\mathfrak{a}$ is the real rank of $G$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "XJS07AJ50Y"
    ],
    "kind": "statement",
    "msc": [
      "22E46",
      "22E15"
    ],
    "name": "Iwasawa decomposition of a Lie group",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q750099",
      "wikipedia": "Iwasawa decomposition"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Iwasawa_decomposition"
      }
    ],
    "tag": "4YDR9Q36DS",
    "uses": [
      "XJS07AJ50Y",
      "A5JB9JQKR0",
      "DK87YN458D",
      "H3QX1ZFC8K"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/4YDR9Q36DS/",
  "links": [
    {
      "field": "about",
      "target": "XJS07AJ50Y"
    },
    {
      "field": "uses",
      "target": "XJS07AJ50Y"
    },
    {
      "field": "uses",
      "target": "A5JB9JQKR0"
    },
    {
      "field": "uses",
      "target": "DK87YN458D"
    },
    {
      "field": "uses",
      "target": "H3QX1ZFC8K"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/4YDR9Q36DS.md",
  "msc": [
    "22E46",
    "22E15"
  ],
  "name": "Iwasawa decomposition of a Lie group",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4YD/4YDR9Q36DS.md",
  "status": null,
  "tag": "4YDR9Q36DS",
  "uri": "m2://4YDR9Q36DS",
  "url": "https://tamnd.github.io/math2/wiki/iwasawa-decomposition-of-a-lie-group/",
  "writing": {
    "tag": "4YDR9Q36DS",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}