example A.358 Rank two: $\mathfrak{su}(3)$

Rests on

• depends_on proposition 14.55 The fundamental, the antifundamental and the adjoint ¶
  • 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 ¶
    • 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 5.147 Invariant subspace ¶
        • depends_on definition 5.7 Vector subspace ¶
        • depends_on definition 5.141 Representation of a 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 ¶
• depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶
  • 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 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 theorem 14.16 Invariant tensors give Casimir operators ¶
      • depends_on definition 14.8 Casimir element ¶ ↺
      • 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 proposition 14.53 Killing form of $\mathfrak{su}(3)$ ¶ ↺
  • 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 ¶
• depends_on theorem A.346 Racah ¶
  • depends_on definition 14.8 Casimir element ¶ ↺
  • depends_on definition 14.20 Cartan subalgebra and rank ¶ ↺
  • depends_on theorem A.330 Cartan's criterion for semisimplicity ¶
    • depends_on definition 14.10 Killing form ¶ ↺
    • depends_on lemma 14.11 Invariance of the Killing form ¶ ↺
    • proves proof app:A-long-proofs@proof-209 ¶
  • proves proof app:A-long-proofs@proof-217 ¶

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