{
  "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"
  },
  "canonicalURL": "https://groups.qft.org/atlas/groups/sl-2-r/",
  "entity": {
    "id": "group:sl-2-r",
    "slug": "sl-2-r",
    "name": "Real special linear group in two dimensions",
    "symbol": "SL(2, R)",
    "category": "lie",
    "level": "reference",
    "aliases": [
      "SU(1,1)"
    ],
    "searchTerms": [
      "sl(2, R)",
      "real rank 1"
    ],
    "tags": [
      "Noncompact Lie group",
      "Simple algebra",
      "Global form"
    ],
    "summary": "Matrix form, a quotient of the universal cover by 2Z. The real 2 × 2 matrices of determinant 1.",
    "definition": "The real 2 × 2 matrices of determinant 1.",
    "definitionEvidence": [
      {
        "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"
      }
    ],
    "properties": [
      {
        "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",
        "evidence": [
          {
            "source": "etingof-real-forms",
            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
          }
        ],
        "assumptions": []
      },
      {
        "key": "center",
        "label": "Center",
        "value": "C₂",
        "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": "pi1",
        "label": "Fundamental group π₁",
        "value": "Z",
        "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": "etingof-global-forms",
            "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
          },
          {
            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
          }
        ],
        "derivation": {
          "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.",
          "inputs": [
            "Maximal compact subgroup: SO(2) ≅ U(1)",
            "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z",
            "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z"
          ]
        },
        "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": [
          {
            "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",
        "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": "Matrix form, a quotient of the universal cover by 2Z",
        "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": "etingof-global-forms",
            "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
          },
          {
            "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"
          }
        ],
        "assumptions": []
      },
      {
        "key": "lie-algebra",
        "label": "Real Lie algebra",
        "value": "sl(2, R)",
        "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"
          }
        ],
        "assumptions": []
      },
      {
        "key": "maximal-compact",
        "label": "Maximal compact subgroup",
        "value": "SO(2) ≅ U(1)",
        "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"
          },
          {
            "source": "etingof-global-forms",
            "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "topology",
        "title": "Global structure and topology",
        "paragraphs": [
          {
            "text": "Polar decomposition gives the manifold SO(2) × R², so the fundamental group is Z. The center is {±I}. The isomorphism with SU(1,1) is an isomorphism of real matrix Lie groups after a complex change of basis.",
            "evidence": [
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
              },
              {
                "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"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "representations",
        "title": "Representation category and descent",
        "paragraphs": [
          {
            "text": "Finite-dimensional complex irreducible representations are restrictions of sl(2,C) highest-weight modules with integer highest weight n ≥ 0 and dimension n + 1. The central element −I acts by (−1)ⁿ. These are distinct from infinite-dimensional unitary representations.",
            "evidence": [
              {
                "source": "conrad",
                "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
              },
              {
                "source": "etingof-lie-notes",
                "locator": "Remark 36.7, printed page 196: finite-dimensional representations of the universal cover factor through SL(2,R)"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-algebra",
        "target": "algebra:sl-2-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: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:sl-2-r-to-psl-2-r",
        "label": "Explicit homomorphism",
        "evidence": [
          {
            "source": "etingof-global-forms",
            "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
          },
          {
            "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-map",
        "target": "map:universal-sl-2-r-to-sl-2-r",
        "label": "Explicit homomorphism",
        "evidence": [
          {
            "source": "etingof-global-forms",
            "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
          },
          {
            "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": "covered-by",
        "target": "group:universal-cover-sl-2-r",
        "label": "Specified connected covering group",
        "evidence": [
          {
            "source": "etingof-global-forms",
            "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
          },
          {
            "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"
          }
        ]
      }
    ],
    "capabilities": [],
    "kind": "group",
    "data": {
      "category": "lie",
      "dimension": 3,
      "compact": false,
      "connected": true,
      "abelian": false,
      "algebraId": "algebra:sl-2-r"
    }
  },
  "claims": [
    {
      "id": "group:sl-2-r#definition",
      "subject": "group:sl-2-r",
      "key": "definition",
      "label": "Definition",
      "value": "The real 2 × 2 matrices of determinant 1.",
      "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"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#dimension",
      "subject": "group:sl-2-r",
      "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": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#real-rank",
      "subject": "group:sl-2-r",
      "key": "real-rank",
      "label": "Real rank",
      "value": "1",
      "evidence": [
        {
          "source": "etingof-real-forms",
          "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#center",
      "subject": "group:sl-2-r",
      "key": "center",
      "label": "Center",
      "value": "C₂",
      "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": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#pi1",
      "subject": "group:sl-2-r",
      "key": "pi1",
      "label": "Fundamental group π₁",
      "value": "Z",
      "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": "etingof-global-forms",
          "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
        },
        {
          "source": "conrad",
          "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
        }
      ],
      "derivation": {
        "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.",
        "inputs": [
          "Maximal compact subgroup: SO(2) ≅ U(1)",
          "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z",
          "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#compact",
      "subject": "group:sl-2-r",
      "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": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#connected",
      "subject": "group:sl-2-r",
      "key": "connected",
      "label": "Connected",
      "value": "Yes",
      "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": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#abelian",
      "subject": "group:sl-2-r",
      "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:sl-2-r#simple-algebra",
      "subject": "group:sl-2-r",
      "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:sl-2-r#global-form",
      "subject": "group:sl-2-r",
      "key": "global-form",
      "label": "Global form",
      "value": "Matrix form, a quotient of the universal cover by 2Z",
      "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": "etingof-global-forms",
          "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
        },
        {
          "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"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#lie-algebra",
      "subject": "group:sl-2-r",
      "key": "lie-algebra",
      "label": "Real Lie algebra",
      "value": "sl(2, R)",
      "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"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#maximal-compact",
      "subject": "group:sl-2-r",
      "key": "maximal-compact",
      "label": "Maximal compact subgroup",
      "value": "SO(2) ≅ U(1)",
      "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"
        },
        {
          "source": "etingof-global-forms",
          "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#topology-1",
      "subject": "group:sl-2-r",
      "key": "topology-1",
      "label": "Global structure and topology",
      "value": "Polar decomposition gives the manifold SO(2) × R², so the fundamental group is Z. The center is {±I}. The isomorphism with SU(1,1) is an isomorphism of real matrix Lie groups after a complex change of basis.",
      "evidence": [
        {
          "source": "etingof-global-forms",
          "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)"
        },
        {
          "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"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#representations-1",
      "subject": "group:sl-2-r",
      "key": "representations-1",
      "label": "Representation category and descent",
      "value": "Finite-dimensional complex irreducible representations are restrictions of sl(2,C) highest-weight modules with integer highest weight n ≥ 0 and dimension n + 1. The central element −I acts by (−1)ⁿ. These are distinct from infinite-dimensional unitary representations.",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lectures 27–28: highest-weight classification and fundamental representations"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Remark 36.7, printed page 196: finite-dimensional representations of the universal cover factor through SL(2,R)"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:sl-2-r#relation-1",
      "subject": "group:sl-2-r",
      "key": "relation-1",
      "label": "Real Lie algebra",
      "value": "algebra:sl-2-r",
      "evidence": [
        {
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