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    "released": "2026-11-08"
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    "symbol": "E₇ compact, simply connected",
    "category": "lie",
    "level": "executable",
    "aliases": [
      "Compact simply connected E7"
    ],
    "searchTerms": [
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    ],
    "tags": [
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    ],
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    "definition": "The connected simply connected compact Lie group integrating the compact real algebra of type E₇.",
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        "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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            "source": "etingof-lie-notes",
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            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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            "source": "etingof-coxeter",
            "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
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        "assumptions": []
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        "key": "center",
        "label": "Center",
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          }
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        "assumptions": []
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        "value": "0",
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            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ],
        "assumptions": []
      },
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        "key": "compact",
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            "source": "conrad",
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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"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
          }
        ],
        "assumptions": []
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        "key": "abelian",
        "label": "Abelian",
        "value": "No",
        "evidence": [
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            "source": "etingof-lie-notes",
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            "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",
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            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          }
        ],
        "assumptions": []
      },
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        "key": "global-form",
        "label": "Global form",
        "value": "Simply connected compact form of E₇",
        "evidence": [
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            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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            "source": "conrad-centers",
            "locator": "Pages 2–4: centers of simply connected groups by root-system type"
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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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        "key": "lie-algebra",
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            "source": "etingof-lie-notes",
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            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
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        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
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        "title": "Global structure and topology",
        "paragraphs": [
          {
            "text": "The center is C₂. Quotienting by it gives the adjoint compact E₇/C₂, with fundamental group C₂.",
            "evidence": [
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                "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": "Dominant integral E₇ highest weights classify finite-dimensional complex irreducibles. An irrep descends to the adjoint group exactly when its central sign is positive, equivalently when its highest weight lies in the root lattice.",
            "evidence": [
              {
                "source": "conrad",
                "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
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                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
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      "compact": true,
      "connected": true,
      "abelian": false,
      "algebraId": "algebra:e-7-compact-r",
      "capabilityPresets": {
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          "labels": "0,0,0,0,0,0,1"
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