{
  "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-so-3/",
  "entity": {
    "id": "map:u-1-to-so-3",
    "slug": "u-1-to-so-3",
    "name": "Normalized axial rotation subgroup of SO(3)",
    "symbol": "U(1) ≅ SO(2) ↪ SO(3)",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Embedding"
    ],
    "summary": "eⁱθ ↦ diag(R(θ),1), where R(θ) has rows (cos θ,−sin θ) and (sin θ,cos θ), with θ modulo 2π. Kernel: {1}.",
    "definition": "eⁱθ ↦ diag(R(θ),1), where R(θ) has rows (cos θ,−sin θ) and (sin θ,cos θ), with θ 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"
      },
      {
        "source": "conrad",
        "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
      }
    ],
    "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"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ],
        "assumptions": []
      },
      {
        "key": "codomain",
        "label": "Codomain",
        "value": "so-3",
        "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": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ],
        "assumptions": []
      },
      {
        "key": "formula",
        "label": "Specified homomorphism",
        "value": "eⁱθ ↦ diag(R(θ),1), where R(θ) has rows (cos θ,−sin θ) and (sin θ,cos θ), with θ 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"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ],
        "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"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ],
        "assumptions": []
      },
      {
        "key": "image",
        "label": "Image",
        "value": "Rotations fixing a chosen oriented axis, isomorphic to SO(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"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ],
        "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"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ],
        "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"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "meaning",
        "title": "Global and representation consequences",
        "paragraphs": [
          {
            "text": "This uses the actual spatial rotation angle θ. The quotient SO(3)/SO(2) is S². The inclusion sends a winding number to its parity in π₁(SO(3)) = C₂.",
            "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": "conrad",
                "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "assumptions": []
          },
          {
            "text": "A 2π rotation lifts to a path from I to −I in SU(2), whereas a 4π rotation lifts to a loop. Therefore the inclusion induces reduction modulo two, and the boundary map π₂(S²) → π₁(SO(2)) is multiplication by +2 after choosing compatible orientations.",
            "evidence": [
              {
                "source": "conrad",
                "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
              },
              {
                "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": "The kernel of Z → C₂ is 2Z. Since π₂(SO(3))=0, the boundary map is injective with this image.",
              "inputs": [
                "map:su-2-to-so-3",
                "SO(3) is RP³",
                "π₂(RP³)=π₂(S³)=0",
                "Fiber loop is one spatial 2π rotation"
              ]
            },
            "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"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ]
      },
      {
        "type": "has-codomain",
        "target": "group:so-3",
        "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"
          },
          {
            "source": "conrad",
            "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
          }
        ]
      },
      {
        "type": "defines-quotient",
        "target": "space:so-3-over-so-2",
        "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:so-3",
      "kernel": "{1}",
      "image": "Rotations fixing a chosen oriented axis, isomorphic to SO(2)",
      "injective": true,
      "surjective": false,
      "capabilityPresets": {
        "topology": {
          "kind": "so3"
        }
      }
    }
  },
  "claims": [
    {
      "id": "map:u-1-to-so-3#definition",
      "subject": "map:u-1-to-so-3",
      "key": "definition",
      "label": "Definition",
      "value": "eⁱθ ↦ diag(R(θ),1), where R(θ) has rows (cos θ,−sin θ) and (sin θ,cos θ), with θ 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"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#domain",
      "subject": "map:u-1-to-so-3",
      "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"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#codomain",
      "subject": "map:u-1-to-so-3",
      "key": "codomain",
      "label": "Codomain",
      "value": "so-3",
      "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": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#formula",
      "subject": "map:u-1-to-so-3",
      "key": "formula",
      "label": "Specified homomorphism",
      "value": "eⁱθ ↦ diag(R(θ),1), where R(θ) has rows (cos θ,−sin θ) and (sin θ,cos θ), with θ 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"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#kernel",
      "subject": "map:u-1-to-so-3",
      "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"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#image",
      "subject": "map:u-1-to-so-3",
      "key": "image",
      "label": "Image",
      "value": "Rotations fixing a chosen oriented axis, isomorphic to SO(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"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#injective",
      "subject": "map:u-1-to-so-3",
      "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"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#surjective",
      "subject": "map:u-1-to-so-3",
      "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"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#meaning-1",
      "subject": "map:u-1-to-so-3",
      "key": "meaning-1",
      "label": "Global and representation consequences",
      "value": "This uses the actual spatial rotation angle θ. The quotient SO(3)/SO(2) is S². The inclusion sends a winding number to its parity in π₁(SO(3)) = C₂.",
      "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": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        },
        {
          "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-so-3#meaning-2",
      "subject": "map:u-1-to-so-3",
      "key": "meaning-2",
      "label": "Global and representation consequences",
      "value": "A 2π rotation lifts to a path from I to −I in SU(2), whereas a 4π rotation lifts to a loop. Therefore the inclusion induces reduction modulo two, and the boundary map π₂(S²) → π₁(SO(2)) is multiplication by +2 after choosing compatible orientations.",
      "evidence": [
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        },
        {
          "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": "The kernel of Z → C₂ is 2Z. Since π₂(SO(3))=0, the boundary map is injective with this image.",
        "inputs": [
          "map:su-2-to-so-3",
          "SO(3) is RP³",
          "π₂(RP³)=π₂(S³)=0",
          "Fiber loop is one spatial 2π rotation"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#relation-1",
      "subject": "map:u-1-to-so-3",
      "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"
        },
        {
          "source": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "relationship": "has-domain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#relation-2",
      "subject": "map:u-1-to-so-3",
      "key": "relation-2",
      "label": "Codomain group",
      "value": "group:so-3",
      "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": "conrad",
          "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3"
        }
      ],
      "assumptions": [],
      "relationship": "has-codomain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:u-1-to-so-3#relation-3",
      "subject": "map:u-1-to-so-3",
      "key": "relation-3",
      "label": "Associated homogeneous quotient",
      "value": "space:so-3-over-so-2",
      "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": [],
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