proposition 14.44 Existence: every allowed $j$ is realized

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1717 · p. 561

Rests on

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 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 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 14.5 Universal enveloping algebra ¶
    - depends_on definition 14.14 Invariant symmetric tensor ¶
      - depends_on equation 14.5 eq:lie-structconst ¶ ↺
    - 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 ¶

Supports

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

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Ladder algebra	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1727
`depends_on`	→	The irreducible representations of $\mathfrak{su}(2)$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1727
`depends_on`	←	Which representations descend to $\SO(3,\R)$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1802
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-18	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1730