{
  "schema": "groups.qft.org/entity/2",
  "versions": {
    "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/maps/su-2-2-to-so-2-4/",
  "entity": {
    "id": "map:su-2-2-to-so-2-4",
    "slug": "su-2-2-to-so-2-4",
    "name": "Exterior-square spin cover in signature (2,4)",
    "symbol": "SU(2,2) → SO₀(2,4)",
    "category": "lie",
    "level": "reference",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Covering or central quotient"
    ],
    "summary": "Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign. Kernel: {±I} ≅ C₂.",
    "definition": "Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.",
    "definitionEvidence": [
      {
        "source": "garrett-isogenies",
        "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
      }
    ],
    "properties": [
      {
        "key": "domain",
        "label": "Domain",
        "value": "su-2-2",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "codomain",
        "label": "Codomain",
        "value": "so-2-4-identity",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "formula",
        "label": "Specified homomorphism",
        "value": "Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "kernel",
        "label": "Kernel",
        "value": "{±I} ≅ C₂",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "image",
        "label": "Image",
        "value": "SO₀(2,4)",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "injective",
        "label": "Injective",
        "value": "No",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "surjective",
        "label": "Surjective",
        "value": "Yes",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
          }
        ],
        "assumptions": []
      },
      {
        "key": "covering-degree",
        "label": "Covering degree",
        "value": "2",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "meaning",
        "title": "Global and representation consequences",
        "paragraphs": [
          {
            "text": "This map identifies SU(2,2) with the connected spin double cover. Its source has π₁ = Z, so this is not a universal covering map. A further quotient of the target by {±I} is a different global group.",
            "evidence": [
              {
                "source": "garrett-isogenies",
                "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-domain",
        "target": "group:su-2-2",
        "label": "Domain group",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
          }
        ]
      },
      {
        "type": "has-codomain",
        "target": "group:so-2-4-identity",
        "label": "Codomain group",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
          }
        ]
      }
    ],
    "capabilities": [],
    "kind": "map",
    "data": {
      "domain": "group:su-2-2",
      "codomain": "group:so-2-4-identity",
      "kernel": "{±I} ≅ C₂",
      "image": "SO₀(2,4)",
      "injective": false,
      "surjective": true
    }
  },
  "claims": [
    {
      "id": "map:su-2-2-to-so-2-4#definition",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "definition",
      "label": "Definition",
      "value": "Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#domain",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "domain",
      "label": "Domain",
      "value": "su-2-2",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#codomain",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "codomain",
      "label": "Codomain",
      "value": "so-2-4-identity",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#formula",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "formula",
      "label": "Specified homomorphism",
      "value": "Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#kernel",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "kernel",
      "label": "Kernel",
      "value": "{±I} ≅ C₂",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#image",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "image",
      "label": "Image",
      "value": "SO₀(2,4)",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#injective",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "injective",
      "label": "Injective",
      "value": "No",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#surjective",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "surjective",
      "label": "Surjective",
      "value": "Yes",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#covering-degree",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "covering-degree",
      "label": "Covering degree",
      "value": "2",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#meaning-1",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "meaning-1",
      "label": "Global and representation consequences",
      "value": "This map identifies SU(2,2) with the connected spin double cover. Its source has π₁ = Z, so this is not a universal covering map. A further quotient of the target by {±I} is a different global group.",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        },
        {
          "source": "etingof-global-forms",
          "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#relation-1",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "relation-1",
      "label": "Domain group",
      "value": "group:su-2-2",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "relationship": "has-domain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-2-2-to-so-2-4#relation-2",
      "subject": "map:su-2-2-to-so-2-4",
      "key": "relation-2",
      "label": "Codomain group",
      "value": "group:so-2-4-identity",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)"
        }
      ],
      "assumptions": [],
      "relationship": "has-codomain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    }
  ],
  "sources": [
    {
      "id": "etingof-global-forms",
      "title": "Lecture 17: Classification of Connected Compact and Complex Reductive Groups",
      "authors": [
        "Pavel Etingof"
      ],
      "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec17.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Sections 43.1–43.2, printed pages 230–235.",
      "licenseNote": "Linked MIT OpenCourseWare notes; global-form and representation criteria are restated."
    },
    {
      "id": "garrett-isogenies",
      "title": "Sporadic isogenies to orthogonal groups",
      "authors": [
        "Paul Garrett"
      ],
      "url": "https://ncatlab.org/nlab/files/Garrett-SporadicIsogenies.pdf",
      "kind": "lecture-notes",
      "locatorHint": "May 7, 2015 notes, Sections 2.1–2.11. The linked file is an accessible mirror of the author’s notes.",
      "licenseNote": "Author’s lecture notes linked at an accessible mirror. No redistribution license is inferred; matrix constructions are explained independently."
    }
  ],
  "review": {
    "status": "pending-expert-review",
    "label": "Expert review pending",
    "currentReviews": 0,
    "staleReviews": 0,
    "fingerprint": "192eedce1f6be85e53247fee72b05d6db8f75fa15ddb0c1fd0b55c84660a8466",
    "reviewers": []
  }
}