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    "name": "Spin cover of three-dimensional rotations",
    "symbol": "SU(2) → SO(3)",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Covering or central quotient"
    ],
    "summary": "A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product. Kernel: {±I} ≅ C₂.",
    "definition": "A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product.",
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        "locator": "Section 2.1: SU(2) → SO(3), construction and kernel"
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          }
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          }
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        "evidence": [
          {
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            "locator": "Section 2.1: SU(2) → SO(3), construction and kernel"
          }
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        "assumptions": []
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            "locator": "Section 2.1: SU(2) → SO(3), construction and kernel"
          }
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        "assumptions": []
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        "title": "Global and representation consequences",
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            "text": "This is the universal covering map. A spin-j SU(2) representation descends along it exactly when 2j is even. With compatible axis conventions, diag(eⁱθ,e⁻ⁱθ) maps to a rotation through 2θ.",
            "evidence": [
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                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
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            "text": "The torus angle in SU(2) has period 2π, but the image rotation has angle 2θ. Tracking that factor is necessary when comparing the two spherical quotient fibrations.",
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            "locator": "Section 2.1: SU(2) → SO(3), construction and kernel"
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      "surjective": true,
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      "key": "covering-degree",
      "label": "Covering degree",
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      "value": "This is the universal covering map. A spin-j SU(2) representation descends along it exactly when 2j is even. With compatible axis conventions, diag(eⁱθ,e⁻ⁱθ) maps to a rotation through 2θ.",
      "evidence": [
        {
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      "subject": "map:su-2-to-so-3",
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      "value": "The torus angle in SU(2) has period 2π, but the image rotation has angle 2θ. Tracking that factor is necessary when comparing the two spherical quotient fibrations.",
      "evidence": [
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          "source": "etingof-lie-notes",
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          "source": "garrett-isogenies",
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  "sources": [
    {
      "id": "etingof-lie-notes",
      "title": "Lie Groups and Lie Algebras",
      "authors": [
        "Pavel Etingof"
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      "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"
      ],
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      "licenseNote": "Linked MIT OpenCourseWare notes; global-form and representation criteria are restated."
    },
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      "title": "Sporadic isogenies to orthogonal groups",
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      "locatorHint": "May 7, 2015 notes, Sections 2.1–2.11. The linked file is an accessible mirror of the author’s notes.",
      "licenseNote": "Author’s lecture notes linked at an accessible mirror. No redistribution license is inferred; matrix constructions are explained independently."
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