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    "level": "reference",
    "aliases": [
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    "searchTerms": [
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    "tags": [
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      "Global form"
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            "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers"
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        "key": "center",
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            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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            "source": "conrad",
            "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights"
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            "source": "conrad",
            "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected"
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        "assumptions": []
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        "key": "abelian",
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        "value": "No",
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            "source": "etingof-lie-notes",
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            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
          }
        ],
        "assumptions": []
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        "key": "simple-algebra",
        "label": "Simple real Lie algebra",
        "value": "No",
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            "source": "etingof-real-forms",
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            "source": "conrad",
            "locator": "Lecture 25 and Appendix T: root systems and classical root computations"
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        "assumptions": []
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            "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras"
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            "source": "conrad",
            "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients"
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            "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375"
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            "locator": "Sections 41.2–41.4: compact, split, and classical real forms"
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        "title": "Global structure and topology",
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          {
            "text": "The surjective map (A,z) ↦ zA from SU(2) × U(1) has kernel {(I,1),(−I,−1)}. The determinant map U(2) → U(1) has fiber SU(2), and induces an isomorphism on fundamental groups.",
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            "text": "Pulling an irrep back to SU(2) × U(1) gives a highest weight n ≥ 0 and an integer U(1) charge q with n + q even. The parity condition makes (−I,−1) act trivially. Its complex dimension is n + 1.",
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