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    "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/universal-cover-sl-2-r/",
  "entity": {
    "id": "group:universal-cover-sl-2-r",
    "slug": "universal-cover-sl-2-r",
    "name": "Simply connected group integrating sl(2,R)",
    "symbol": "Universal cover of SL(2, R)",
    "category": "lie",
    "level": "reference",
    "aliases": [
      "Universal cover of PSL(2,R)",
      "Universal cover of SU(1,1)"
    ],
    "searchTerms": [
      "sl(2, R)",
      "real rank 1"
    ],
    "tags": [
      "Noncompact Lie group",
      "Simple algebra",
      "Simply connected"
    ],
    "summary": "Simply connected real Lie group; not a matrix group. The simply connected covering Lie group of SL(2,R), or equivalently of PSL(2,R).",
    "definition": "The simply connected covering Lie group of SL(2,R), or equivalently of PSL(2,R).",
    "definitionEvidence": [
      {
        "source": "brodzki-sl2-cover",
        "locator": "Section 1, page 2: the universal covering group of SL(2,R)"
      }
    ],
    "properties": [
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        "key": "dimension",
        "label": "Real dimension",
        "value": "3",
        "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"
          },
          {
            "source": "etingof-coxeter",
            "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
          }
        ],
        "assumptions": []
      },
      {
        "key": "real-rank",
        "label": "Real rank",
        "value": "1",
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            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ],
        "assumptions": []
      },
      {
        "key": "center",
        "label": "Center",
        "value": "Z",
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          {
            "source": "brodzki-sl2-cover",
            "locator": "Section 1, page 2: infinite cyclic center and absence of nontrivial compact subgroups"
          }
        ],
        "assumptions": []
      },
      {
        "key": "pi1",
        "label": "Fundamental group π₁",
        "value": "0",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Section 3.2: universal covering Lie groups"
          }
        ],
        "assumptions": []
      },
      {
        "key": "compact",
        "label": "Compact",
        "value": "No",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "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"
          }
        ],
        "assumptions": []
      },
      {
        "key": "connected",
        "label": "Connected",
        "value": "Yes",
        "evidence": [
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            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "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"
          }
        ],
        "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"
          },
          {
            "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",
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            "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 real Lie group; not a matrix group",
        "evidence": [
          {
            "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"
          },
          {
            "source": "brodzki-sl2-cover",
            "locator": "Section 1, page 2: structure of the universal cover"
          },
          {
            "source": "knapp-structure",
            "locator": "Theorem 6.46: Iwasawa decomposition"
          },
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          }
        ],
        "assumptions": []
      },
      {
        "key": "lie-algebra",
        "label": "Real Lie algebra",
        "value": "sl(2, R)",
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            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
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            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ],
        "assumptions": []
      },
      {
        "key": "maximal-compact",
        "label": "Maximal compact subgroup",
        "value": "Trivial",
        "evidence": [
          {
            "source": "brodzki-sl2-cover",
            "locator": "Section 1, page 2: infinite cyclic center and absence of nontrivial compact subgroups"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "topology",
        "title": "Global structure and topology",
        "paragraphs": [
          {
            "text": "The lifted Iwasawa factors identify its manifold with R³. Let z be the central element represented by a lift of a rotation through π in SL(2,R). Then the projection to SL(2,R) has kernel ⟨z²⟩, while the projection to PSL(2,R) has kernel ⟨z⟩.",
            "evidence": [
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              },
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                "source": "knapp-structure",
                "locator": "Theorem 6.46: Iwasawa decomposition"
              },
              {
                "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": "Every finite-dimensional complex representation factors through SL(2,R). Consequently this simply connected Lie group has no faithful finite-dimensional matrix representation; its infinite cyclic center is not detected faithfully in that category.",
            "evidence": [
              {
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                "locator": "Remark 36.7, printed page 196; Exercise 11.20: finite-dimensional representations and nonlinearity of the universal cover"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
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        "target": "algebra:sl-2-r",
        "label": "Real Lie algebra",
        "evidence": [
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          },
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ]
      },
      {
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        "target": "real-form:sl-2-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:universal-sl-2-r-to-sl-2-r",
        "label": "Explicit homomorphism",
        "evidence": [
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            "source": "brodzki-sl2-cover",
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            "source": "knapp-structure",
            "locator": "Theorem 6.46: Iwasawa decomposition"
          },
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          }
        ]
      }
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    "capabilities": [],
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    "data": {
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      "compact": false,
      "connected": true,
      "abelian": false,
      "algebraId": "algebra:sl-2-r"
    }
  },
  "claims": [
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      "subject": "group:universal-cover-sl-2-r",
      "key": "definition",
      "label": "Definition",
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    },
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      "id": "group:universal-cover-sl-2-r#connected",
      "subject": "group:universal-cover-sl-2-r",
      "key": "connected",
      "label": "Connected",
      "value": "Yes",
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          "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"
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    },
    {
      "id": "group:universal-cover-sl-2-r#abelian",
      "subject": "group:universal-cover-sl-2-r",
      "key": "abelian",
      "label": "Abelian",
      "value": "No",
      "evidence": [
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          "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"
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      "subject": "group:universal-cover-sl-2-r",
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      "label": "Simple real Lie algebra",
      "value": "Yes",
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          "source": "conrad",
          "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:universal-cover-sl-2-r#global-form",
      "subject": "group:universal-cover-sl-2-r",
      "key": "global-form",
      "label": "Global form",
      "value": "Simply connected real Lie group; not a matrix group",
      "evidence": [
        {
          "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"
        },
        {
          "source": "brodzki-sl2-cover",
          "locator": "Section 1, page 2: structure of the universal cover"
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        {
          "source": "knapp-structure",
          "locator": "Theorem 6.46: Iwasawa decomposition"
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          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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    },
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      "subject": "group:universal-cover-sl-2-r",
      "key": "lie-algebra",
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      "key": "maximal-compact",
      "label": "Maximal compact subgroup",
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      "assumptions": [],
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    },
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      "value": "The lifted Iwasawa factors identify its manifold with R³. Let z be the central element represented by a lift of a rotation through π in SL(2,R). Then the projection to SL(2,R) has kernel ⟨z²⟩, while the projection to PSL(2,R) has kernel ⟨z⟩.",
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    },
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      "subject": "group:universal-cover-sl-2-r",
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      "label": "Representation category and descent",
      "value": "Every finite-dimensional complex representation factors through SL(2,R). Consequently this simply connected Lie group has no faithful finite-dimensional matrix representation; its infinite cyclic center is not detected faithfully in that category.",
      "evidence": [
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          "source": "etingof-lie-notes",
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      "label": "Explicit homomorphism",
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          "locator": "Section 1, page 2: structure of the universal cover"
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          "source": "knapp-structure",
          "locator": "Theorem 6.46: Iwasawa decomposition"
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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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      "id": "conrad",
      "title": "Conrad and Landesman · Compact Lie groups",
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        "Aaron Landesman"
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      "title": "Lie Groups and Lie Algebras",
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    {
      "id": "brodzki-sl2-cover",
      "title": "K-theory and limit-of-discrete-series for the universal cover of SL₂(R)",
      "authors": [
        "Jacek Brodzki",
        "Graham Niblo",
        "Roger Plymen",
        "Nick Wright"
      ],
      "url": "https://eprints.maths.manchester.ac.uk/2082/1/Dirac_soton_v14.pdf",
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      "locatorHint": "Section 1, page 2, describes the universal covering group and its center.",
      "licenseNote": "Linked institutional-repository preprint. Only introductory structural facts are summarized."
    }
  ],
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