lemma 14.41 Ladder algebra

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1591 · p. 559

Rests on

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 ¶
      - step_from equation 14.4 eq:lie-gengrlie ¶
  - 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 equation 14.5 eq:lie-structconst ¶ ↺
  - 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 ¶

Supports

depends_on proposition 14.44 Existence: every allowed $j$ is realized ¶
- depends_on proposition 14.45 Which representations descend to $\SO(3,\R)$ ¶
depends_on theorem 14.43 The irreducible representations of $\mathfrak{su}(2)$ ¶
- depends_on example A.394 $D = 4$ ¶
- depends_on example A.357 Rank one: $\mathfrak{su}(2)$ ¶
- depends_on proposition 14.44 Existence: every allowed $j$ is realized ¶ ↺
- depends_on theorem 14.47 Clebsch–Gordan series ¶

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`	→	The quadratic Casimir	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1606
`depends_on`	→	Ladder operators and the Casimir	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1606
`depends_on`	→	eq:lie-killing-components	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1606
`depends_on`	→	eq:lie-su2-algebra	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1606
`depends_on`	←	Existence: every allowed $j$ is realized	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:1659
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-16	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1609