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  "entity": {
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    "slug": "lie-c",
    "name": "Simple Lie type Cᵣ",
    "symbol": "Cᵣ",
    "category": "lie",
    "level": "classification",
    "aliases": [],
    "searchTerms": [
      "Cr",
      "Dynkin classification",
      "simple Lie algebra",
      "Sp(r), compact quaternionic convention"
    ],
    "tags": [
      "Classification",
      "Complex simple algebra",
      "Compact global forms"
    ],
    "summary": "Complex simple Lie type Cᵣ with dimension, normalized Coxeter data, compact cover, and its center.",
    "definition": "sp(2r, C), with integer r ≥ 3 in the nonredundant classification.",
    "definitionEvidence": [
      {
        "source": "conrad",
        "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
      },
      {
        "source": "etingof-real-forms",
        "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
      }
    ],
    "properties": [
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        "key": "rank",
        "label": "Rank",
        "value": "integer r ≥ 3 in the nonredundant classification",
        "evidence": [
          {
            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          }
        ],
        "assumptions": []
      },
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        "key": "dimension",
        "label": "Complex algebra dimension",
        "value": "r(2r + 1)",
        "evidence": [
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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"
          }
        ],
        "derivation": {
          "method": "For an irreducible root system, dim(g) = r + |Φ| = r(h + 1).",
          "inputs": [
            "Rank r",
            "Coxeter number h in Section 32.4"
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        },
        "assumptions": []
      },
      {
        "key": "coxeter-number",
        "label": "Coxeter number h",
        "value": "2r",
        "evidence": [
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            "source": "etingof-coxeter",
            "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
          }
        ],
        "assumptions": []
      },
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        "key": "dual-coxeter-number",
        "label": "Dual Coxeter number h∨",
        "value": "r + 1",
        "evidence": [
          {
            "source": "etingof-coxeter",
            "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
          }
        ],
        "assumptions": []
      },
      {
        "key": "compact-cover",
        "label": "Compact simply connected group",
        "value": "Sp(r), compact quaternionic convention",
        "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": "center",
        "label": "Center of compact simply connected group",
        "value": "C₂",
        "evidence": [
          {
            "source": "conrad-centers",
            "locator": "Pages 2–4: centers of simply connected groups by root-system type"
          }
        ],
        "assumptions": []
      },
      {
        "key": "root-normalization",
        "label": "Root-length convention",
        "value": "Long roots have squared length 2",
        "evidence": [
          {
            "source": "etingof-coxeter",
            "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "objects",
        "title": "Algebra, real form, global group",
        "paragraphs": [
          {
            "text": "A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.",
            "evidence": [
              {
                "source": "etingof-real-forms",
                "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
              },
              {
                "source": "conrad",
                "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
              }
            ],
            "assumptions": []
          },
          {
            "text": "Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.",
            "evidence": [
              {
                "source": "conrad",
                "locator": "Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "representations",
        "title": "Representation convention",
        "paragraphs": [
          {
            "text": "Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.",
            "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"
              }
            ],
            "assumptions": []
          },
          {
            "text": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.",
            "evidence": [
              {
                "source": "etingof-coxeter",
                "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "real-forms",
        "title": "Classical real forms",
        "paragraphs": [
          {
            "text": "The real forms are sp(2r,R) and sp(p,q), p + q = r. Here sp(p,q) preserves a quaternionic Hermitian form, whereas sp(2r,R) acts on a real vector space of dimension 2r.",
            "evidence": [
              {
                "source": "etingof-real-forms",
                "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
              },
              {
                "source": "knapp-structure",
                "locator": "Theorem 6.105, printed page 362: nonredundant real-form classification and coincidences"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "low-rank",
        "title": "Rank conventions and coincidences",
        "paragraphs": [
          {
            "text": "r = 1: C₁ = A₁.",
            "evidence": [
              {
                "source": "conrad-centers",
                "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
              },
              {
                "source": "conrad",
                "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality"
              }
            ],
            "assumptions": []
          },
          {
            "text": "r = 2: C₂ = B₂; see the canonical B₂ algebra and compact Sp(2) group records.",
            "evidence": [
              {
                "source": "conrad-centers",
                "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms"
              },
              {
                "source": "conrad",
                "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
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        "label": "Concrete algebra record",
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            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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    "capabilities": [],
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          "at": "r = 1",
          "description": "C₁ = A₁."
        },
        {
          "at": "r = 2",
          "description": "C₂ = B₂; see the canonical B₂ algebra and compact Sp(2) group records."
        }
      ],
      "scope": "Classification of complex simple type; compact center and global-form rule; selected concrete real forms."
    }
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      "key": "coxeter-number",
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          "source": "etingof-coxeter",
          "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
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    },
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      "label": "Dual Coxeter number h∨",
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    },
    {
      "id": "family:lie-c#compact-cover",
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      "key": "compact-cover",
      "label": "Compact simply connected group",
      "value": "Sp(r), compact quaternionic convention",
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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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      "label": "Center of compact simply connected group",
      "value": "C₂",
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          "locator": "Pages 2–4: centers of simply connected groups by root-system type"
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      "subject": "family:lie-c",
      "key": "root-normalization",
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      "value": "Long roots have squared length 2",
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          "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
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      "assumptions": [],
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    {
      "id": "family:lie-c#objects-1",
      "subject": "family:lie-c",
      "key": "objects-1",
      "label": "Algebra, real form, global group",
      "value": "A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.",
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          "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"
    },
    {
      "id": "family:lie-c#objects-2",
      "subject": "family:lie-c",
      "key": "objects-2",
      "label": "Algebra, real form, global group",
      "value": "Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.",
      "evidence": [
        {
          "source": "conrad",
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      "assumptions": [],
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    },
    {
      "id": "family:lie-c#representations-2",
      "subject": "family:lie-c",
      "key": "representations-2",
      "label": "Representation convention",
      "value": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.",
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    },
    {
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      "key": "real-forms-1",
      "label": "Classical real forms",
      "value": "The real forms are sp(2r,R) and sp(p,q), p + q = r. Here sp(p,q) preserves a quaternionic Hermitian form, whereas sp(2r,R) acts on a real vector space of dimension 2r.",
      "evidence": [
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          "source": "etingof-real-forms",
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          "locator": "Theorem 6.105, printed page 362: nonredundant real-form classification and coincidences"
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      "label": "Rank conventions and coincidences",
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          "source": "conrad-centers",
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          "source": "conrad",
          "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality"
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      "reviewStatus": "pending-expert-review",
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    },
    {
      "id": "family:lie-c#low-rank-2",
      "subject": "family:lie-c",
      "key": "low-rank-2",
      "label": "Rank conventions and coincidences",
      "value": "r = 2: C₂ = B₂; see the canonical B₂ algebra and compact Sp(2) group records.",
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          "source": "conrad",
          "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality"
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      "title": "Conrad and Landesman · Compact Lie groups",
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        "Aaron Landesman"
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      "id": "etingof-coxeter",
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      "authors": [
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      "locatorHint": "Sections 32.1–32.5, printed pages 172–178.",
      "licenseNote": "Linked MIT OpenCourseWare notes. Numerical root-system facts are used with original explanations."
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      "id": "etingof-real-forms",
      "title": "Lecture 15: Classification of Real Forms of Semisimple Lie Algebras",
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      "kind": "lecture-notes",
      "locatorHint": "Section 41, printed pages 218–225.",
      "licenseNote": "Linked MIT OpenCourseWare notes; definitions and classification facts are summarized in original prose."
    },
    {
      "id": "etingof-global-forms",
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        "Pavel Etingof"
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      "id": "knapp-structure",
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      "locatorHint": "Chapter VI, including Theorem 6.46 and Theorem 6.105 (printed page 362).",
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