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    "released": "2026-11-08"
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    "slug": "universal-sl-2-r-to-sl-2-r",
    "name": "Universal covering map of SL(2,R)",
    "symbol": "Universal SL(2,R) cover → SL(2,R)",
    "category": "lie",
    "level": "reference",
    "aliases": [],
    "searchTerms": [],
    "tags": [
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    ],
    "summary": "Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism. Kernel: ⟨z²⟩ ≅ Z, where z lifts a π rotation.",
    "definition": "Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism.",
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            "locator": "Section 1, page 2: universal cover and its infinite cyclic center"
          }
        ],
        "assumptions": []
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          }
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          }
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          }
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          }
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        "assumptions": []
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          }
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          }
        ],
        "assumptions": []
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        "value": "Infinitely many sheets",
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          }
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        "assumptions": []
      }
    ],
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      {
        "id": "meaning",
        "title": "Global and representation consequences",
        "paragraphs": [
          {
            "text": "The center of the domain is ⟨z⟩. Its subgroup ⟨z²⟩ is the kernel of this projection; composing with SL(2,R) → PSL(2,R) enlarges the kernel to the full center.",
            "evidence": [
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              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "assumptions": []
          },
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            "text": "Choose R(θ) as the standard planar rotation matrix inside SL(2,R). The path θ:0→π ends at −I and defines z; the path θ:0→2π is one fundamental loop in SL(2,R) and defines z².",
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                "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
              }
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                "R(π)=−I"
              ]
            },
            "assumptions": []
          }
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      "value": "The center of the domain is ⟨z⟩. Its subgroup ⟨z²⟩ is the kernel of this projection; composing with SL(2,R) → PSL(2,R) enlarges the kernel to the full center.",
      "evidence": [
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      "value": "Choose R(θ) as the standard planar rotation matrix inside SL(2,R). The path θ:0→π ends at −I and defines z; the path θ:0→2π is one fundamental loop in SL(2,R) and defines z².",
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        "Jacek Brodzki",
        "Graham Niblo",
        "Roger Plymen",
        "Nick Wright"
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      "url": "https://eprints.maths.manchester.ac.uk/2082/1/Dirac_soton_v14.pdf",
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