{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "1NX95TTMFT"
    },
    {
      "field": "about",
      "target": "4YDR9Q36DS"
    },
    {
      "field": "about",
      "target": "A5JB9JQKR0"
    },
    {
      "field": "about",
      "target": "C34RZMKKA4"
    },
    {
      "field": "about",
      "target": "C846XS3Y0T"
    },
    {
      "field": "about",
      "target": "PG1CVVN7KA"
    },
    {
      "field": "uses",
      "target": "4YDR9Q36DS"
    },
    {
      "field": "uses",
      "target": "A5JB9JQKR0"
    },
    {
      "field": "uses",
      "target": "BA2S40MM0Y"
    },
    {
      "field": "uses",
      "target": "BM3TWJ0D4P"
    },
    {
      "field": "uses",
      "target": "C34RZMKKA4"
    },
    {
      "field": "uses",
      "target": "C846XS3Y0T"
    },
    {
      "field": "uses",
      "target": "DYXEBVDC6W"
    },
    {
      "field": "uses",
      "target": "FEJK846N5X"
    },
    {
      "field": "uses",
      "target": "JC9R1073M9"
    },
    {
      "field": "uses",
      "target": "K10YADF5XV"
    },
    {
      "field": "uses",
      "target": "M9DY1E7RKS"
    },
    {
      "field": "uses",
      "target": "PG1CVVN7KA"
    },
    {
      "field": "uses",
      "target": "SP421Y498C"
    },
    {
      "field": "uses",
      "target": "T5KZR54SG8"
    },
    {
      "field": "uses",
      "target": "TM8PYTBT6Y"
    },
    {
      "field": "uses",
      "target": "V2HTMG4KT1"
    },
    {
      "field": "specializes",
      "target": "AM7YJP8AR6"
    }
  ],
  "body": "\n## Summary\n\nA semisimple Lie group is a connected [[Lie group|7JBRGCVJRB]] whose Lie algebra is a semisimple Lie algebra, that is, has no nonzero solvable ideals.\nEquivalently, its Lie algebra is a direct sum of simple Lie algebras.\nSo, up to a covering and a central quotient, the group is a product of simple Lie groups.\nSemisimple Lie groups have a rich structure theory, with Cartan, Iwasawa and Bruhat decompositions, and a deep representation theory.\nBy the Levi decomposition, a finite-dimensional real or complex Lie algebra is a semisimple subalgebra plus its largest solvable ideal.\nSo semisimple and solvable groups are the two basic kinds.\n\n## Explanations\n\n### L1 human\n\nA semisimple Lie group is a Lie group built from simple Lie groups, with no commutative or \"almost commutative\" continuous part.\nThe group $SO(4)$ is a typical example.\nIts Lie algebra splits into two copies of the Lie algebra of $SO(3)$, one for each of two independent kinds of rotation of four-dimensional space.\nThe unitary group $U(n)$ is not semisimple, because the scalar matrices $e^{i\\theta} I$ form a commutative continuous part.\nRemoving that part by fixing the determinant gives the semisimple group $SU(n)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22E46",
      "22E15",
      "22E10"
    ],
    "name": "Semisimple Lie group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Simple_Lie_group"
      }
    ],
    "specializes": [
      "42ZCE14AD2"
    ],
    "tag": "XJS07AJ50Y"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/XJS07AJ50Y/",
  "links": [
    {
      "field": "specializes",
      "target": "42ZCE14AD2"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/XJS07AJ50Y.md",
  "msc": [
    "22E46",
    "22E15",
    "22E10"
  ],
  "name": "Semisimple Lie group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XJS/XJS07AJ50Y.md",
  "status": null,
  "tag": "XJS07AJ50Y",
  "uri": "m2://XJS07AJ50Y",
  "url": "https://tamnd.github.io/math2/wiki/semisimple-lie-group/",
  "writing": {
    "tag": "XJS07AJ50Y",
    "kind": "concept",
    "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
      }
    ]
  }
}