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  "entity": {
    "id": "algebra:standard-model-r",
    "slug": "standard-model-r",
    "name": "Compact Standard Model gauge algebra",
    "symbol": "su(3) ⊕ su(2) ⊕ u(1)",
    "category": "lie",
    "level": "reference",
    "aliases": [
      "Standard Model algebra"
    ],
    "searchTerms": [
      "Standard Model",
      "hypercharge",
      "reductive",
      "semisimple part"
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    "tags": [
      "Real algebra",
      "Compact",
      "Reductive"
    ],
    "summary": "The twelve-dimensional compact reductive real algebra su(3) ⊕ su(2) ⊕ u(1), shared by four specified matter-compatible compact global groups.",
    "definition": "The direct sum of the compact real algebras su(3), su(2), and u(1). Its real dimension is 8+3+1=12, its rank is 2+1+1=4, and its one-dimensional center makes it reductive but not semisimple.",
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        "source": "tong",
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      },
      {
        "source": "conrad-pi1-lie",
        "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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    "properties": [
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            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
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          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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        "key": "rank",
        "label": "Rank",
        "value": "4",
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            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
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          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
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          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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        "assumptions": []
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      {
        "key": "center",
        "label": "Lie-algebra center",
        "value": "u(1), dimension one",
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            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
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          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
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            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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        "assumptions": []
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        "key": "simple",
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            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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        "assumptions": []
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    "sections": [
      {
        "id": "global-forms",
        "title": "Global groups and charge normalization",
        "paragraphs": [
          {
            "text": "The Lie algebra alone does not fix a connected global gauge group or the allowed line operators. The four group records here use the compact product SU(3) × SU(2) × U(1), integer charge Q=6Y, and subgroups of its specified diagonal Z6.",
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                "source": "tong",
                "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
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                "source": "conrad-pi1-lie",
                "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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            "source": "tong",
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            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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        ]
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            "source": "tong",
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            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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        "type": "integrates-to",
        "target": "group:standard-model-z2",
        "label": "Compact global form with quotient order 2",
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            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
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            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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        "type": "integrates-to",
        "target": "group:standard-model-z3",
        "label": "Compact global form with quotient order 3",
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            "source": "block-baez-huerta",
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          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
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            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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        "label": "Compact global form with quotient order 6",
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            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
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          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
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          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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    "capabilities": [],
    "kind": "algebra",
    "data": {
      "field": "R",
      "dimension": 12,
      "rank": 4,
      "simple": false,
      "displayTex": "\\mathfrak{su}(3)\\oplus\\mathfrak{su}(2)\\oplus\\mathfrak u(1)"
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  "claims": [
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          "source": "tong",
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          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
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          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
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        {
          "source": "tong",
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          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
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      "id": "tong",
      "title": "Tong · Line operators in the Standard Model",
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      "title": "The Algebra of Grand Unified Theories",
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      "id": "conrad-pi1-lie",
      "title": "Brian Conrad, Fundamental Groups of Lie Groups",
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        "Brian Conrad"
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