theorem 14.13 Cartan's criterion

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:437 · p. 546

Rests on

depends_on definition 14.10 Killing form ¶
- depends_on equation 14.5 eq:lie-structconst ¶
  - step_from equation 14.4 eq:lie-gengrlie ¶
    - step_from equation 14.3 eq:lie-generators ¶
    - step_from equation 14.2 eq:lie-varincor ¶
proves proof ch:12-lie-groups-fibre-bundles@prooflink-2 ¶

Supports

depends_on corollary 14.17 The quadratic Casimir ¶
- depends_on lemma 14.41 Ladder algebra ¶
  - 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 ¶
- depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶
  - depends_on example A.358 Rank two: $\mathfrak{su}(3)$ ¶
depends_on lemma A.427 Invariant bilinear forms on a simple algebra ¶
- depends_on proposition A.428 Uniqueness in degree zero ¶
depends_on proposition 14.53 Killing form of $\mathfrak{su}(3)$ ¶
- depends_on proposition 14.55 The fundamental, the antifundamental and the adjoint ¶
  - depends_on corollary A.377 The adjoint representation is irreducible ¶
  - depends_on example A.358 Rank two: $\mathfrak{su}(3)$ ¶ ↺
  - depends_on theorem 14.57 $\vect{3}\otimes\bar{\vect{3}} = \vect{1}\oplus\vect{8}$ ¶
    - depends_on proposition 102.6 Mesons are $q\bar{q}$, baryons are $qqq$ ¶
      - assumes experiment Deep Inelastic Scattering ¶
      - depends_on phenomenon 116.38 The quark spins do not account for the proton spin ¶
      - depends_on phenomenon 102.9 The $\Delta^{++}$ violates the spin-statistics theorem ¶
      - depends_on phenomenon 102.7 The nucleon magnetic moments come out right ¶
      - depends_on proposition 116.29 The Gross–Llewellyn Smith sum rule in the parton model ¶
  - depends_on theorem 102.12 Which quark combinations can be colour singlets ¶
    - depends_on corollary 102.13 The selection rule of the eightfold way ¶
- depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶ ↺
depends_on proposition 14.77 Semisimple algebras admit no nontrivial extension ¶
- depends_on proposition A.428 Uniqueness in degree zero ¶ ↺
- depends_on proposition 14.84 Properties of a contraction ¶

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`	→	Killing form	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:442
`depends_on`	←	The quadratic Casimir	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:529
`depends_on`	←	Invariant bilinear forms on a simple algebra	declared	appendices/A-long-proofs.tex:21002
`depends_on`	←	Killing form of $\mathfrak{su}(3)$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2073
`depends_on`	←	Semisimple algebras admit no nontrivial extension	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3496
`proves`	←	ch:12-lie-groups-fibre-bundles@prooflink-2	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:451