{
  "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/maps/u-1-to-su-2/",
  "entity": {
    "id": "map:u-1-to-su-2",
    "slug": "u-1-to-su-2",
    "name": "Normalized maximal torus of SU(2)",
    "symbol": "U(1) ↪ SU(2)",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Embedding"
    ],
    "summary": "eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π. Kernel: {1}.",
    "definition": "eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π.",
    "definitionEvidence": [
      {
        "source": "etingof-lie-notes",
        "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
      },
      {
        "source": "etingof-lie-notes",
        "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
      }
    ],
    "properties": [
      {
        "key": "domain",
        "label": "Domain",
        "value": "u-1",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ],
        "assumptions": []
      },
      {
        "key": "codomain",
        "label": "Codomain",
        "value": "su-2",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ],
        "assumptions": []
      },
      {
        "key": "formula",
        "label": "Specified homomorphism",
        "value": "eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π.",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ],
        "assumptions": []
      },
      {
        "key": "kernel",
        "label": "Kernel",
        "value": "{1}",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ],
        "assumptions": []
      },
      {
        "key": "image",
        "label": "Image",
        "value": "{diag(z,z⁻¹) : |z|=1}",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ],
        "assumptions": []
      },
      {
        "key": "injective",
        "label": "Injective",
        "value": "Yes",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ],
        "assumptions": []
      },
      {
        "key": "surjective",
        "label": "Surjective",
        "value": "No",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "meaning",
        "title": "Global and representation consequences",
        "paragraphs": [
          {
            "text": "The defining doublet has U(1) weights +1 and −1 in this convention. Its homogeneous quotient is SU(2)/U(1) ≅ S².",
            "evidence": [
              {
                "source": "etingof-lie-notes",
                "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
              },
              {
                "source": "etingof-lie-notes",
                "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
              },
              {
                "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 induced map π₁(U(1)) → π₁(SU(2)) is zero. In the quotient fibration, the boundary map π₂(S²) → π₁(U(1)) is multiplication by +1 when the sphere generator is oriented to match this fiber generator.",
            "evidence": [
              {
                "source": "etingof-lie-notes",
                "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
              },
              {
                "source": "hatcher-algebraic-topology",
                "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
              }
            ],
            "derivation": {
              "method": "Apply the exact sequence to U(1) → SU(2) → S², using π₂(S³)=π₁(S³)=0.",
              "inputs": [
                "SU(2) is S³",
                "π₁(U(1))=Z",
                "Fiber loop θ:0→2π",
                "Choice of positively compatible sphere generator"
              ]
            },
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-domain",
        "target": "group:u-1",
        "label": "Domain group",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ]
      },
      {
        "type": "has-codomain",
        "target": "group:su-2",
        "label": "Codomain group",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
          },
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ]
      },
      {
        "type": "defines-quotient",
        "target": "space:su-2-over-u-1",
        "label": "Associated homogeneous quotient",
        "evidence": [
          {
            "source": "etingof-lie-notes",
            "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
          }
        ]
      }
    ],
    "capabilities": [
      "topology"
    ],
    "kind": "map",
    "data": {
      "domain": "group:u-1",
      "codomain": "group:su-2",
      "kernel": "{1}",
      "image": "{diag(z,z⁻¹) : |z|=1}",
      "injective": true,
      "surjective": false,
      "capabilityPresets": {
        "topology": {
          "kind": "su2"
        }
      }
    }
  },
  "claims": [
    {
      "id": "map:u-1-to-su-2#definition",
      "subject": "map:u-1-to-su-2",
      "key": "definition",
      "label": "Definition",
      "value": "eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#domain",
      "subject": "map:u-1-to-su-2",
      "key": "domain",
      "label": "Domain",
      "value": "u-1",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#codomain",
      "subject": "map:u-1-to-su-2",
      "key": "codomain",
      "label": "Codomain",
      "value": "su-2",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#formula",
      "subject": "map:u-1-to-su-2",
      "key": "formula",
      "label": "Specified homomorphism",
      "value": "eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#kernel",
      "subject": "map:u-1-to-su-2",
      "key": "kernel",
      "label": "Kernel",
      "value": "{1}",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#image",
      "subject": "map:u-1-to-su-2",
      "key": "image",
      "label": "Image",
      "value": "{diag(z,z⁻¹) : |z|=1}",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#injective",
      "subject": "map:u-1-to-su-2",
      "key": "injective",
      "label": "Injective",
      "value": "Yes",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#surjective",
      "subject": "map:u-1-to-su-2",
      "key": "surjective",
      "label": "Surjective",
      "value": "No",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#meaning-1",
      "subject": "map:u-1-to-su-2",
      "key": "meaning-1",
      "label": "Global and representation consequences",
      "value": "The defining doublet has U(1) weights +1 and −1 in this convention. Its homogeneous quotient is SU(2)/U(1) ≅ S².",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        },
        {
          "source": "etingof-global-forms",
          "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#meaning-2",
      "subject": "map:u-1-to-su-2",
      "key": "meaning-2",
      "label": "Global and representation consequences",
      "value": "The induced map π₁(U(1)) → π₁(SU(2)) is zero. In the quotient fibration, the boundary map π₂(S²) → π₁(U(1)) is multiplication by +1 when the sphere generator is oriented to match this fiber generator.",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        },
        {
          "source": "hatcher-algebraic-topology",
          "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
        }
      ],
      "derivation": {
        "method": "Apply the exact sequence to U(1) → SU(2) → S², using π₂(S³)=π₁(S³)=0.",
        "inputs": [
          "SU(2) is S³",
          "π₁(U(1))=Z",
          "Fiber loop θ:0→2π",
          "Choice of positively compatible sphere generator"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#relation-1",
      "subject": "map:u-1-to-su-2",
      "key": "relation-1",
      "label": "Domain group",
      "value": "group:u-1",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "relationship": "has-domain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#relation-2",
      "subject": "map:u-1-to-su-2",
      "key": "relation-2",
      "label": "Codomain group",
      "value": "group:su-2",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
        },
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "relationship": "has-codomain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-su-2#relation-3",
      "subject": "map:u-1-to-su-2",
      "key": "relation-3",
      "label": "Associated homogeneous quotient",
      "value": "space:su-2-over-u-1",
      "evidence": [
        {
          "source": "etingof-lie-notes",
          "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³"
        }
      ],
      "assumptions": [],
      "relationship": "defines-quotient",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    }
  ],
  "sources": [
    {
      "id": "etingof-lie-notes",
      "title": "Lie Groups and Lie Algebras",
      "authors": [
        "Pavel Etingof"
      ],
      "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf",
      "kind": "lecture-notes",
      "locatorHint": "MIT 18.755, Spring 2024. Cite section numbers and printed pages.",
      "licenseNote": "Linked MIT OpenCourseWare lecture notes. Original summaries only; website content is not offered as a reproduction of these notes."
    },
    {
      "id": "etingof-global-forms",
      "title": "Lecture 17: Classification of Connected Compact and Complex Reductive Groups",
      "authors": [
        "Pavel Etingof"
      ],
      "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec17.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Sections 43.1–43.2, printed pages 230–235.",
      "licenseNote": "Linked MIT OpenCourseWare notes; global-form and representation criteria are restated."
    },
    {
      "id": "hatcher-algebraic-topology",
      "title": "Algebraic Topology",
      "authors": [
        "Allen Hatcher"
      ],
      "url": "https://pi.math.cornell.edu/~hatcher/AT/AT.pdf",
      "kind": "book",
      "locatorHint": "Section 4.2, Theorem 4.41 (printed page 376) and the preceding Hopf-fibration example.",
      "licenseNote": "Linked author-hosted textbook. Exact sequences below are independently derived from the stated theorem."
    }
  ],
  "review": {
    "status": "pending-expert-review",
    "label": "Expert review pending",
    "currentReviews": 0,
    "staleReviews": 0,
    "fingerprint": "47c90ee3748b4bb364100525ca7631ffe974f8e0ed76471243fdc4f581525378",
    "reviewers": []
  }
}