example A.357 Rank one: $\mathfrak{su}(2)$

open in the book · appendices/A-long-proofs.tex:17251 · p. 2963

example -- no derivation owed by its kind

Rests on

depends_on theorem A.346 Racah ¶
- depends_on definition 14.8 Casimir element ¶
  - depends_on definition 14.5 Universal enveloping algebra ¶
    - depends_on equation 14.5 eq:lie-structconst ¶
      - step_from equation 14.4 eq:lie-gengrlie ¶
- depends_on definition 14.20 Cartan subalgebra and rank ¶
- depends_on theorem A.330 Cartan's criterion for semisimplicity ¶
  - depends_on definition 14.10 Killing form ¶
    - depends_on equation 14.5 eq:lie-structconst ¶ ↺
  - depends_on lemma 14.11 Invariance of the Killing form ¶
    - depends_on definition 14.10 Killing form ¶ ↺
    - proves proof ch:12-lie-groups-fibre-bundles@proof-2 ¶
  - proves proof app:A-long-proofs@proof-209 ¶
- proves proof app:A-long-proofs@proof-217 ¶
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 ¶
  - 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 ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-17 ¶

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type	direction	node	provenance	where
`depends_on`	→	Racah	declared	appendices/A-long-proofs.tex:17262
`depends_on`	→	The irreducible representations of $\mathfrak{su}(2)$	declared	appendices/A-long-proofs.tex:17262