{
  "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/sp-4-r-to-so-2-3/",
  "entity": {
    "id": "map:sp-4-r-to-so-2-3",
    "slug": "sp-4-r-to-so-2-3",
    "name": "Primitive-exterior-square spin cover in signature (2,3)",
    "symbol": "Sp(4, R) → SO₀(2,3)",
    "category": "lie",
    "level": "reference",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Covering or central quotient"
    ],
    "summary": "For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign. Kernel: {±I} ≅ C₂.",
    "definition": "For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.",
    "definitionEvidence": [
      {
        "source": "garrett-isogenies",
        "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
      },
      {
        "source": "etingof-lie-notes",
        "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
      }
    ],
    "properties": [
      {
        "key": "domain",
        "label": "Domain",
        "value": "sp-4-r",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
          }
        ],
        "assumptions": []
      },
      {
        "key": "codomain",
        "label": "Codomain",
        "value": "so-2-3-identity",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
          }
        ],
        "assumptions": []
      },
      {
        "key": "formula",
        "label": "Specified homomorphism",
        "value": "For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
          }
        ],
        "assumptions": []
      },
      {
        "key": "kernel",
        "label": "Kernel",
        "value": "{±I} ≅ C₂",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
          }
        ],
        "assumptions": []
      },
      {
        "key": "image",
        "label": "Image",
        "value": "SO₀(2,3)",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
          }
        ],
        "assumptions": []
      },
      {
        "key": "injective",
        "label": "Injective",
        "value": "No",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
          }
        ],
        "assumptions": []
      },
      {
        "key": "surjective",
        "label": "Surjective",
        "value": "Yes",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
          }
        ],
        "assumptions": []
      },
      {
        "key": "covering-degree",
        "label": "Covering degree",
        "value": "2",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "meaning",
        "title": "Global and representation consequences",
        "paragraphs": [
          {
            "text": "The source has maximal compact U(2) and fundamental group Z; the target has maximal compact SO(2) × SO(3) and fundamental group Z × C₂. The covering is double but is not universal.",
            "evidence": [
              {
                "source": "garrett-isogenies",
                "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
              },
              {
                "source": "etingof-lie-notes",
                "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "assumptions": []
          },
          {
            "text": "On maximal compact subgroups, the covering may be written U(2) → SO(2) × SO(3) using the determinant and the projective SU(2) action. A generator of π₁(U(2)) maps to (1,1 mod 2), an index-two subgroup of Z × C₂.",
            "evidence": [
              {
                "source": "etingof-lie-notes",
                "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
              },
              {
                "source": "conrad",
                "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
              },
              {
                "source": "knapp-structure",
                "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316"
              }
            ],
            "derivation": {
              "method": "Represent a generator by diag(eⁱᵗ,1), 0≤t≤2π. Its determinant winds once and its SU(2) lift ends at −I, giving the nontrivial SO(3) loop.",
              "inputs": [
                "U(2)=(SU(2)×U(1))/diagonal C₂",
                "SU(2)→SO(3) has kernel {±I}",
                "The determinant map induces π₁(U(2))≅Z"
              ]
            },
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-domain",
        "target": "group:sp-4-r",
        "label": "Domain group",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
          }
        ]
      },
      {
        "type": "has-codomain",
        "target": "group:so-2-3-identity",
        "label": "Codomain group",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
          }
        ]
      }
    ],
    "capabilities": [],
    "kind": "map",
    "data": {
      "domain": "group:sp-4-r",
      "codomain": "group:so-2-3-identity",
      "kernel": "{±I} ≅ C₂",
      "image": "SO₀(2,3)",
      "injective": false,
      "surjective": true
    }
  },
  "claims": [
    {
      "id": "map:sp-4-r-to-so-2-3#definition",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "definition",
      "label": "Definition",
      "value": "For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#domain",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "domain",
      "label": "Domain",
      "value": "sp-4-r",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#codomain",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "codomain",
      "label": "Codomain",
      "value": "so-2-3-identity",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#formula",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "formula",
      "label": "Specified homomorphism",
      "value": "For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#kernel",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "kernel",
      "label": "Kernel",
      "value": "{±I} ≅ C₂",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#image",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "image",
      "label": "Image",
      "value": "SO₀(2,3)",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#injective",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "injective",
      "label": "Injective",
      "value": "No",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#surjective",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "surjective",
      "label": "Surjective",
      "value": "Yes",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#covering-degree",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "covering-degree",
      "label": "Covering degree",
      "value": "2",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#meaning-1",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "meaning-1",
      "label": "Global and representation consequences",
      "value": "The source has maximal compact U(2) and fundamental group Z; the target has maximal compact SO(2) × SO(3) and fundamental group Z × C₂. The covering is double but is not universal.",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        },
        {
          "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:sp-4-r-to-so-2-3#meaning-2",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "meaning-2",
      "label": "Global and representation consequences",
      "value": "On maximal compact subgroups, the covering may be written U(2) → SO(2) × SO(3) using the determinant and the projective SU(2) action. A generator of π₁(U(2)) maps to (1,1 mod 2), an index-two subgroup of Z × C₂.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        },
        {
          "source": "knapp-structure",
          "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316"
        }
      ],
      "derivation": {
        "method": "Represent a generator by diag(eⁱᵗ,1), 0≤t≤2π. Its determinant winds once and its SU(2) lift ends at −I, giving the nontrivial SO(3) loop.",
        "inputs": [
          "U(2)=(SU(2)×U(1))/diagonal C₂",
          "SU(2)→SO(3) has kernel {±I}",
          "The determinant map induces π₁(U(2))≅Z"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#relation-1",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "relation-1",
      "label": "Domain group",
      "value": "group:sp-4-r",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "relationship": "has-domain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:sp-4-r-to-so-2-3#relation-2",
      "subject": "map:sp-4-r-to-so-2-3",
      "key": "relation-2",
      "label": "Codomain group",
      "value": "group:so-2-3-identity",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module"
        }
      ],
      "assumptions": [],
      "relationship": "has-codomain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    }
  ],
  "sources": [
    {
      "id": "conrad",
      "title": "Conrad and Landesman · Compact Lie groups",
      "authors": [
        "Brian Conrad",
        "Aaron Landesman"
      ],
      "url": "https://math.stanford.edu/~conrad/210CPage/handouts/lie_groups_notes.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Numbered lectures and appendices; page numbers refer to the printed pagination.",
      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "etingof-lie-notes",
      "title": "Lie Groups and Lie Algebras",
      "authors": [
        "Pavel Etingof"
      ],
      "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf",
      "kind": "lecture-notes",
      "locatorHint": "MIT 18.755, Spring 2024. Cite section numbers and printed pages.",
      "licenseNote": "Linked MIT OpenCourseWare lecture notes. Original summaries only; website content is not offered as a reproduction of these notes."
    },
    {
      "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": "knapp-structure",
      "title": "Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple Groups",
      "authors": [
        "Anthony W. Knapp"
      ],
      "url": "https://www.math.stonybrook.edu/~aknapp/books/green/file4.pdf",
      "kind": "book",
      "locatorHint": "Chapter VI, including Theorem 6.46 and Theorem 6.105 (printed page 362).",
      "licenseNote": "Linked author-hosted chapter excerpt. No reproduction or redistribution permission is assumed."
    },
    {
      "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": "9746c5b45365acee2bf2a0534fde7910eb1837d5a7e2dfa905769fac732dca05",
    "reviewers": []
  }
}