proposition 14.55 The fundamental, the antifundamental and the adjoint

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2207 · p. 566

Rests on

depends_on corollary 14.18 A Casimir acts as a number on an irreducible representation ¶
- 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 equation 14.6 eq:lie-expgen ¶
- depends_on remark 14.6 What $U(\mathfrak{g})$ is for ¶
- depends_on theorem 5.158 Schur's first lemma ¶
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.147 Invariant subspace ¶
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶
    - depends_on definition 5.141 Representation of a group ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.43 Group homomorphism ¶
      - depends_on definition 4.48 Symmetric group ¶
  - depends_on definition 5.148 Irreducible representation ¶
    - depends_on definition 5.147 Invariant subspace ¶ ↺
  - proves proof ch:03-linear-algebra-representations@proof-67 ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-6 ¶
depends_on definition 14.51 Gell-Mann basis ¶
- depends_on definition 14.49 The special unitary group in three dimensions ¶
depends_on proposition 14.53 Killing form of $\mathfrak{su}(3)$ ¶
- depends_on definition 14.51 Gell-Mann basis ¶ ↺
- depends_on equation 14.15 eq:lie-killing-components ¶
- depends_on proposition 14.52 Structure constants and the symmetric tensor ¶
  - depends_on definition 14.51 Gell-Mann basis ¶ ↺
  - proves proof ch:12-lie-groups-fibre-bundles@proof-22 ¶
  - proves proof ch:12-lie-groups-fibre-bundles@prooflink-5 ¶
- depends_on theorem 14.13 Cartan's criterion ¶
  - depends_on definition 14.10 Killing form ¶
    - depends_on equation 14.5 eq:lie-structconst ¶ ↺
  - proves proof ch:12-lie-groups-fibre-bundles@prooflink-2 ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-23 ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-25 ¶

Supports

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 ¶
    - cites (data) derived bib 1 Combined measurement and QCD analysis of the inclusive e $\pm$ p scattering cross sections at HERA ¶
    - cites (data) derived bib 1 Combined measurement and QCD analysis of the inclusive e $\pm$ p scattering cross sections at HERA ¶ ↺
    - cites (evidence) derived bib 1 Combined measurement and QCD analysis of the inclusive e $\pm$ p scattering cross sections at HERA ¶ ↺
    - cites (data) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶
    - cites (data) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶ ↺
    - cites (data) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶ ↺
    - cites (data) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶ ↺
    - cites (data) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶ ↺
    - cites (evidence) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶ ↺
    - cites (context) derived bib 9 Electron-Ion Collider: The next QCD frontier ¶
    - cites (context) derived bib 9 Electron-Ion Collider: The next QCD frontier ¶ ↺
    - cites (context) derived bib 2 Asymptotic freedom in parton language ¶
    - … 102 more
  - 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 ¶

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`	→	A Casimir acts as a number on an irreducible representation	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2227
`depends_on`	→	Gell-Mann basis	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2227
`depends_on`	→	Killing form of $\mathfrak{su}(3)$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2227
`depends_on`	←	The adjoint representation is irreducible	declared	appendices/A-long-proofs.tex:18438
`depends_on`	←	Rank two: $\mathfrak{su}(3)$	declared	appendices/A-long-proofs.tex:17278
`depends_on`	←	$\vect{3}\otimes\bar{\vect{3}} = \vect{1}\oplus\vect{8}$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2289
`depends_on`	←	Which quark combinations can be colour singlets	declared	parts/11-qft-standard-model/04-qcd.tex:508
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-25	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2230