proposition 14.45 Which representations descend to $\SO(3,\R)$

Rests on

• depends_on corollary 14.38 $\SU(2)$ is the universal cover ¶
  • depends_on proposition 14.33 $\SU(2)$ is the three-sphere ¶
    • depends_on definition 14.32 The special unitary group in two dimensions ¶
    • proves proof ch:12-lie-groups-fibre-bundles@proof-11 ¶
  • depends_on theorem 14.37 $\SU(2)$ is a two-to-one cover of $\SO(3,\R)$ ¶
    • depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶
      • depends_on equation 14.28 eq:lie-expso2 ¶
      • depends_on equation 14.25 eq:lie-so2-matrix ¶
      • depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
        • depends_on definition 14.26 The rotation group ¶
        • proves proof ch:12-lie-groups-fibre-bundles@proof-8 ¶
      • proves proof ch:12-lie-groups-fibre-bundles@proof-9 ¶
    • depends_on proposition 14.36 Closed form of the exponential ¶
      • depends_on equation 14.28 eq:lie-expso2 ¶ ↺
      • depends_on proposition 14.35 Generators of $\mathfrak{su}(2)$ ¶
        • depends_on definition 14.32 The special unitary group in two dimensions ¶ ↺
        • proves proof ch:12-lie-groups-fibre-bundles@proof-13 ¶
      • proves proof ch:12-lie-groups-fibre-bundles@proof-14 ¶
    • depends_on proposition 14.35 Generators of $\mathfrak{su}(2)$ ¶ ↺
    • depends_on proposition 14.33 $\SU(2)$ is the three-sphere ¶ ↺
    • proves proof ch:12-lie-groups-fibre-bundles@proof-15 ¶
  • proves proof app:A-long-proofs@proof-225 ¶
  • proves proof ch:12-lie-groups-fibre-bundles@prooflink-4 ¶
• depends_on proposition 14.44 Existence: every allowed $j$ is realized ¶
  • depends_on lemma 14.41 Ladder algebra ¶
    • depends_on corollary 14.17 The quadratic Casimir ¶
      • depends_on corollary 14.12 Total antisymmetry of the structure constants ¶
        • depends_on definition 14.10 Killing form ¶
        • depends_on lemma 14.11 Invariance of the Killing form ¶
        • proves proof ch:12-lie-groups-fibre-bundles@proof-3 ¶
      • depends_on theorem 14.13 Cartan's criterion ¶
        • depends_on definition 14.10 Killing form ¶ ↺
        • proves proof ch:12-lie-groups-fibre-bundles@prooflink-2 ¶
      • depends_on theorem 14.16 Invariant tensors give Casimir operators ¶
        • depends_on definition 14.8 Casimir element ¶
        • depends_on definition 14.14 Invariant symmetric tensor ¶
        • depends_on definition 14.5 Universal enveloping algebra ¶
        • proves proof ch:12-lie-groups-fibre-bundles@proof-4 ¶
      • proves proof ch:12-lie-groups-fibre-bundles@proof-5 ¶
    • depends_on definition 14.40 Ladder operators and the Casimir ¶
      • depends_on equation 14.44 eq:lie-su2-algebra ¶
    • depends_on equation 14.15 eq:lie-killing-components ¶
    • depends_on equation 14.44 eq:lie-su2-algebra ¶ ↺
    • proves proof ch:12-lie-groups-fibre-bundles@proof-16 ¶
  • depends_on theorem 14.43 The irreducible representations of $\mathfrak{su}(2)$ ¶
    • depends_on equation 14.44 eq:lie-su2-algebra ¶ ↺
    • depends_on lemma 14.41 Ladder algebra ¶ ↺
    • proves proof ch:12-lie-groups-fibre-bundles@proof-17 ¶
  • proves proof ch:12-lie-groups-fibre-bundles@proof-18 ¶
• depends_on proposition 14.36 Closed form of the exponential ¶ ↺
• proves proof ch:12-lie-groups-fibre-bundles@proof-19 ¶

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