proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2173 · p. 566

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 proposition 14.50 Dimension and rank ¶
- depends_on definition 14.20 Cartan subalgebra and rank ¶
- depends_on definition 14.49 The special unitary group in three dimensions ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-21 ¶
depends_on proposition 14.52 Structure constants and the symmetric tensor ¶
- depends_on definition 14.51 Gell-Mann basis ¶
  - depends_on definition 14.49 The special unitary group in three dimensions ¶ ↺
- proves proof ch:12-lie-groups-fibre-bundles@proof-22 ¶
- proves proof ch:12-lie-groups-fibre-bundles@prooflink-5 ¶
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 theorem 14.13 Cartan's criterion ¶ ↺
- proves proof ch:12-lie-groups-fibre-bundles@proof-23 ¶
depends_on theorem 14.16 Invariant tensors give Casimir operators ¶ ↺
depends_on theorem 14.110 thm:lie-invpoly ¶
- depends_on definition 14.109 $G$-invariant polynomial ¶
  - depends_on equation 14.144 eq:lie-invpol-linear ¶
  - depends_on equation 14.145 eq:lie-invpol-symm ¶
- depends_on equation 14.5 eq:lie-structconst ¶ ↺
- proves proof ch:12-lie-groups-fibre-bundles@proof-45 ¶
depends_on theorem 14.21 Racah ¶
- depends_on definition 14.8 Casimir element ¶ ↺
- depends_on definition 14.20 Cartan subalgebra and rank ¶ ↺
- proves proof ch:12-lie-groups-fibre-bundles@prooflink-3 ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-24 ¶

Supports

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`	→	The quadratic Casimir	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2182
`depends_on`	→	Dimension and rank	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2182
`depends_on`	→	Structure constants and the symmetric tensor	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2182
`depends_on`	→	Killing form of $\mathfrak{su}(3)$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2182
`depends_on`	→	Invariant tensors give Casimir operators	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2182
`depends_on`	→	thm:lie-invpoly	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2182
`depends_on`	→	Racah	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2182
`depends_on`	←	Rank two: $\mathfrak{su}(3)$	declared	appendices/A-long-proofs.tex:17278
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-24	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2185