{
  "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"
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  "canonicalURL": "https://groups.qft.org/atlas/groups/sp-2/",
  "entity": {
    "id": "group:sp-2",
    "slug": "sp-2",
    "name": "Compact symplectic group of quaternionic rank two",
    "symbol": "Sp(2)",
    "category": "lie",
    "level": "executable",
    "aliases": [
      "USp(4)",
      "Spin(5)"
    ],
    "searchTerms": [
      "sp(2) ≅ so(5)"
    ],
    "tags": [
      "Compact Lie group",
      "Simple algebra",
      "Simply connected"
    ],
    "summary": "Simply connected compact form of B₂ = C₂. Quaternionic unitary 2 × 2 matrices. Equivalently, this is the compact subgroup of U(4) preserving a complex symplectic form.",
    "definition": "Quaternionic unitary 2 × 2 matrices. Equivalently, this is the compact subgroup of U(4) preserving a complex symplectic form.",
    "definitionEvidence": [
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        "source": "etingof-lie-notes",
        "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
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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"
      }
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    "properties": [
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        "key": "dimension",
        "label": "Real dimension",
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        "evidence": [
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            "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": []
      },
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        "key": "rank",
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        "evidence": [
          {
            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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        ],
        "assumptions": []
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        "key": "center",
        "label": "Center",
        "value": "C₂",
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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",
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        "evidence": [
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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            "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",
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        "evidence": [
          {
            "source": "conrad",
            "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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            "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"
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          {
            "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"
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          {
            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          }
        ],
        "assumptions": []
      },
      {
        "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": "Simply connected compact form of B₂ = C₂",
        "evidence": [
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          },
          {
            "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"
          },
          {
            "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"
          }
        ],
        "assumptions": []
      },
      {
        "key": "lie-algebra",
        "label": "Real Lie algebra",
        "value": "sp(2) ≅ so(5)",
        "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": "Sp(2) is compact and simply connected, with center {±I}. The isomorphism Spin(5) ≅ Sp(2) is an isomorphism of global groups, not only of Lie algebras.",
            "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"
              },
              {
                "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"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "representations",
        "title": "Representation category and descent",
        "paragraphs": [
          {
            "text": "The defining complex 4 preserves a nondegenerate antisymmetric form and has quaternionic type. The vector 5 is the traceless part of the exterior square of this 4; it factors through SO(5).",
            "evidence": [
              {
                "source": "conrad",
                "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
              },
              {
                "source": "garrett-isogenies",
                "locator": "Section 2.6: five-dimensional orthogonal action of compact Sp(2)"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "notation",
        "title": "Symplectic notation",
        "paragraphs": [
          {
            "text": "This site writes compact Sp(n) for quaternionic n × n unitary matrices, also called USp(2n). It writes Sp(2n,R) for real matrices of size 2n preserving an alternating form. The two conventions count different dimensions.",
            "evidence": [
              {
                "source": "etingof-lie-notes",
                "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-algebra",
        "target": "algebra:sp-2-compact-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:sp-2-compact-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:sp-2-to-so-5",
        "label": "Explicit homomorphism",
        "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"
          },
          {
            "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"
          }
        ]
      }
    ],
    "capabilities": [
      "lie-representations"
    ],
    "kind": "group",
    "data": {
      "category": "lie",
      "dimension": 10,
      "compact": true,
      "connected": true,
      "abelian": false,
      "algebraId": "algebra:sp-2-compact-r",
      "capabilityPresets": {
        "lie-representations": {
          "type": "C",
          "rank": "2",
          "labels": "1,0"
        }
      }
    }
  },
  "claims": [
    {
      "id": "group:sp-2#definition",
      "subject": "group:sp-2",
      "key": "definition",
      "label": "Definition",
      "value": "Quaternionic unitary 2 × 2 matrices. Equivalently, this is the compact subgroup of U(4) preserving a complex symplectic form.",
      "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": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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    },
    {
      "id": "group:sp-2#dimension",
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          "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": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sp-2#rank",
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      "key": "rank",
      "label": "Compact rank",
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        {
          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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      ],
      "assumptions": [],
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      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sp-2#center",
      "subject": "group:sp-2",
      "key": "center",
      "label": "Center",
      "value": "C₂",
      "evidence": [
        {
          "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"
        }
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      "assumptions": [],
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      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sp-2#pi1",
      "subject": "group:sp-2",
      "key": "pi1",
      "label": "Fundamental group π₁",
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      "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:sp-2#compact",
      "subject": "group:sp-2",
      "key": "compact",
      "label": "Compact",
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      "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"
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      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sp-2#connected",
      "subject": "group:sp-2",
      "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:sp-2#abelian",
      "subject": "group:sp-2",
      "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:sp-2#simple-algebra",
      "subject": "group:sp-2",
      "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:sp-2#global-form",
      "subject": "group:sp-2",
      "key": "global-form",
      "label": "Global form",
      "value": "Simply connected compact form of B₂ = C₂",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        },
        {
          "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"
        },
        {
          "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"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sp-2#lie-algebra",
      "subject": "group:sp-2",
      "key": "lie-algebra",
      "label": "Real Lie algebra",
      "value": "sp(2) ≅ so(5)",
      "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"
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      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sp-2#topology-1",
      "subject": "group:sp-2",
      "key": "topology-1",
      "label": "Global structure and topology",
      "value": "Sp(2) is compact and simply connected, with center {±I}. The isomorphism Spin(5) ≅ Sp(2) is an isomorphism of global groups, not only of Lie algebras.",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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        {
          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
        },
        {
          "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"
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      "revision": "2026.11-workbench.4"
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      "id": "group:sp-2#representations-1",
      "subject": "group:sp-2",
      "key": "representations-1",
      "label": "Representation category and descent",
      "value": "The defining complex 4 preserves a nondegenerate antisymmetric form and has quaternionic type. The vector 5 is the traceless part of the exterior square of this 4; it factors through SO(5).",
      "evidence": [
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          "source": "conrad",
          "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
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        {
          "source": "garrett-isogenies",
          "locator": "Section 2.6: five-dimensional orthogonal action of compact Sp(2)"
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      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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      "id": "group:sp-2#notation-1",
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      "key": "relation-1",
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          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
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      "relationship": "has-algebra",
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      "id": "group:sp-2#relation-2",
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      "key": "relation-2",
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          "source": "etingof-real-forms",
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      "relationship": "has-real-form",
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      "id": "group:sp-2#relation-3",
      "subject": "group:sp-2",
      "key": "relation-3",
      "label": "Explicit homomorphism",
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          "source": "conrad",
          "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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          "source": "conrad",
          "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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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"
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      "title": "Conrad and Landesman · Compact Lie groups",
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      "id": "etingof-lie-notes",
      "title": "Lie Groups and Lie Algebras",
      "authors": [
        "Pavel Etingof"
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      "title": "Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual Representations",
      "authors": [
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    {
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      "title": "Sporadic isogenies to orthogonal groups",
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