proposition 14.53 Killing form of $\mathfrak{su}(3)$

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2064 · p. 564

Rests on

depends_on definition 14.51 Gell-Mann basis ¶
- depends_on definition 14.49 The special unitary group in three dimensions ¶
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 ¶
    - 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 ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-23 ¶

Supports

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 ¶
      - 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 ¶
depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶
- depends_on example A.358 Rank two: $\mathfrak{su}(3)$ ¶ ↺

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`	→	Gell-Mann basis	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2073
`depends_on`	→	eq:lie-killing-components	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2073
`depends_on`	→	Structure constants and the symmetric tensor	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2073
`depends_on`	→	Cartan's criterion	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2073
`depends_on`	←	The fundamental, the antifundamental and the adjoint	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2227
`depends_on`	←	The two Casimir operators of $\mathfrak{su}(3)$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2182
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-23	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2076