{
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    "schema": "2.0.0",
    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
  "canonicalURL": "https://groups.qft.org/atlas/groups/psu-4/",
  "entity": {
    "id": "group:psu-4",
    "slug": "psu-4",
    "name": "Adjoint compact group of type A₃",
    "symbol": "PSU(4)",
    "category": "lie",
    "level": "reference",
    "aliases": [
      "SU(4)/C4",
      "PSO(6)"
    ],
    "searchTerms": [
      "su(4) ≅ so(6)"
    ],
    "tags": [
      "Compact Lie group",
      "Simple algebra",
      "Global form"
    ],
    "summary": "SU(4)/C₄. The quotient of SU(4) by its full scalar center C₄.",
    "definition": "The quotient of SU(4) by its full scalar center C₄.",
    "definitionEvidence": [
      {
        "source": "etingof-lie-notes",
        "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
      },
      {
        "source": "conrad",
        "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
      },
      {
        "source": "conrad",
        "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
      }
    ],
    "properties": [
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        "key": "dimension",
        "label": "Real dimension",
        "value": "15",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
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          {
            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          },
          {
            "source": "etingof-coxeter",
            "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
          }
        ],
        "assumptions": []
      },
      {
        "key": "rank",
        "label": "Compact rank",
        "value": "3",
        "evidence": [
          {
            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          }
        ],
        "assumptions": []
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      {
        "key": "center",
        "label": "Center",
        "value": "1",
        "evidence": [
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            "source": "conrad-centers",
            "locator": "Pages 2–4: centers of simply connected groups by root-system type"
          },
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          }
        ],
        "assumptions": []
      },
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        "key": "pi1",
        "label": "Fundamental group π₁",
        "value": "C₄",
        "evidence": [
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ],
        "assumptions": []
      },
      {
        "key": "compact",
        "label": "Compact",
        "value": "Yes",
        "evidence": [
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            "source": "conrad",
            "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
          }
        ],
        "assumptions": []
      },
      {
        "key": "connected",
        "label": "Connected",
        "value": "Yes",
        "evidence": [
          {
            "source": "conrad",
            "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
          }
        ],
        "assumptions": []
      },
      {
        "key": "abelian",
        "label": "Abelian",
        "value": "No",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          }
        ],
        "assumptions": []
      },
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        "key": "simple-algebra",
        "label": "Simple real Lie algebra",
        "value": "Yes",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          },
          {
            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          }
        ],
        "assumptions": []
      },
      {
        "key": "global-form",
        "label": "Global form",
        "value": "SU(4)/C₄",
        "evidence": [
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          },
          {
            "source": "conrad-centers",
            "locator": "Pages 2–4: centers of simply connected groups by root-system type"
          },
          {
            "source": "conrad-centers",
            "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
          },
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          }
        ],
        "assumptions": []
      },
      {
        "key": "lie-algebra",
        "label": "Real Lie algebra",
        "value": "su(4) ≅ so(6)",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
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          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "topology",
        "title": "Global structure and topology",
        "paragraphs": [
          {
            "text": "Its simply connected cover is SU(4), with covering kernel C₄. Through Spin(6) ≅ SU(4), this quotient is PSO(6) = SO(6)/{±I}.",
            "evidence": [
              {
                "source": "conrad-centers",
                "locator": "Pages 2–4: centers of simply connected groups by root-system type"
              },
              {
                "source": "conrad-centers",
                "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
              },
              {
                "source": "conrad",
                "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "representations",
        "title": "Representation category and descent",
        "paragraphs": [
          {
            "text": "An SU(4) highest weight with Dynkin labels (a,b,c) descends exactly when a + 2b + 3c is divisible by 4. This criterion concerns the center action, not the reality type.",
            "evidence": [
              {
                "source": "conrad",
                "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "derivation": {
              "method": "The k-th exterior-power fundamental representation has scalar-center charge k; add charges with the Dynkin-label multiplicities.",
              "inputs": [
                "The center generator iI of SU(4)",
                "Fundamental weights realized by Λ¹, Λ², and Λ³ of C⁴"
              ]
            },
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-algebra",
        "target": "algebra:su-4-r",
        "label": "Real Lie algebra",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ]
      },
      {
        "type": "has-real-form",
        "target": "real-form:su-4-r",
        "label": "Real-form context",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ]
      },
      {
        "type": "has-map",
        "target": "map:su-4-to-psu-4",
        "label": "Explicit homomorphism",
        "evidence": [
          {
            "source": "conrad-centers",
            "locator": "Pages 2–4: centers of simply connected groups by root-system type"
          },
          {
            "source": "conrad-centers",
            "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
          },
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          }
        ]
      },
      {
        "type": "covered-by",
        "target": "group:su-4",
        "label": "Specified connected covering group",
        "evidence": [
          {
            "source": "conrad-centers",
            "locator": "Pages 2–4: centers of simply connected groups by root-system type"
          },
          {
            "source": "conrad-centers",
            "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
          },
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          }
        ]
      }
    ],
    "capabilities": [],
    "kind": "group",
    "data": {
      "category": "lie",
      "dimension": 15,
      "compact": true,
      "connected": true,
      "abelian": false,
      "algebraId": "algebra:su-4-r"
    }
  },
  "claims": [
    {
      "id": "group:psu-4#definition",
      "subject": "group:psu-4",
      "key": "definition",
      "label": "Definition",
      "value": "The quotient of SU(4) by its full scalar center C₄.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "conrad",
          "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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      "revision": "2026.11-workbench.4"
    },
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      "id": "group:psu-4#dimension",
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          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
        },
        {
          "source": "etingof-coxeter",
          "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#rank",
      "subject": "group:psu-4",
      "key": "rank",
      "label": "Compact rank",
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      "evidence": [
        {
          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#center",
      "subject": "group:psu-4",
      "key": "center",
      "label": "Center",
      "value": "1",
      "evidence": [
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          "source": "conrad-centers",
          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#pi1",
      "subject": "group:psu-4",
      "key": "pi1",
      "label": "Fundamental group π₁",
      "value": "C₄",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#compact",
      "subject": "group:psu-4",
      "key": "compact",
      "label": "Compact",
      "value": "Yes",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#connected",
      "subject": "group:psu-4",
      "key": "connected",
      "label": "Connected",
      "value": "Yes",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#abelian",
      "subject": "group:psu-4",
      "key": "abelian",
      "label": "Abelian",
      "value": "No",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#simple-algebra",
      "subject": "group:psu-4",
      "key": "simple-algebra",
      "label": "Simple real Lie algebra",
      "value": "Yes",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        },
        {
          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#global-form",
      "subject": "group:psu-4",
      "key": "global-form",
      "label": "Global form",
      "value": "SU(4)/C₄",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
        },
        {
          "source": "conrad-centers",
          "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#lie-algebra",
      "subject": "group:psu-4",
      "key": "lie-algebra",
      "label": "Real Lie algebra",
      "value": "su(4) ≅ so(6)",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#topology-1",
      "subject": "group:psu-4",
      "key": "topology-1",
      "label": "Global structure and topology",
      "value": "Its simply connected cover is SU(4), with covering kernel C₄. Through Spin(6) ≅ SU(4), this quotient is PSO(6) = SO(6)/{±I}.",
      "evidence": [
        {
          "source": "conrad-centers",
          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
        },
        {
          "source": "conrad-centers",
          "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#representations-1",
      "subject": "group:psu-4",
      "key": "representations-1",
      "label": "Representation category and descent",
      "value": "An SU(4) highest weight with Dynkin labels (a,b,c) descends exactly when a + 2b + 3c is divisible by 4. This criterion concerns the center action, not the reality type.",
      "evidence": [
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          "source": "etingof-global-forms",
          "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
        }
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      "derivation": {
        "method": "The k-th exterior-power fundamental representation has scalar-center charge k; add charges with the Dynkin-label multiplicities.",
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        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:psu-4#relation-1",
      "subject": "group:psu-4",
      "key": "relation-1",
      "label": "Real Lie algebra",
      "value": "algebra:su-4-r",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
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        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        }
      ],
      "assumptions": [],
      "relationship": "has-algebra",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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    {
      "id": "group:psu-4#relation-2",
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      "relationship": "has-real-form",
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      "key": "relation-3",
      "label": "Explicit homomorphism",
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      "evidence": [
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          "source": "conrad-centers",
          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
        },
        {
          "source": "conrad-centers",
          "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        }
      ],
      "assumptions": [],
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      "revision": "2026.11-workbench.4"
    },
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      "id": "group:psu-4#relation-4",
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      "key": "relation-4",
      "label": "Specified connected covering group",
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          "source": "conrad-centers",
          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
        },
        {
          "source": "conrad-centers",
          "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        }
      ],
      "assumptions": [],
      "relationship": "covered-by",
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      "title": "Conrad and Landesman · Compact Lie groups",
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        "Aaron Landesman"
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    },
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      "licenseNote": "Linked author-hosted handout; factual center data are restated in original notation."
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      "id": "etingof-lie-notes",
      "title": "Lie Groups and Lie Algebras",
      "authors": [
        "Pavel Etingof"
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