proposition A.428 Uniqueness in degree zero

Rests on

• depends_on lemma A.427 Invariant bilinear forms on a simple algebra ¶
  • depends_on lemma 14.11 Invariance of the Killing form ¶
    • depends_on definition 14.10 Killing form ¶
      • depends_on equation 14.5 eq:lie-structconst ¶
        • step_from equation 14.4 eq:lie-gengrlie ¶
    • proves proof ch:12-lie-groups-fibre-bundles@proof-2 ¶
  • 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 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 app:A-long-proofs@proof-257 ¶
• depends_on proposition 14.77 Semisimple algebras admit no nontrivial extension ¶
  • depends_on definition 14.75 The classifying group ¶
    • depends_on definition 14.74 Coboundary and triviality ¶
      • depends_on proposition 14.73 The extension datum is a $2$-cocycle ¶
        • depends_on definition 14.72 Central extension ¶
        • proves proof ch:12-lie-groups-fibre-bundles@proof-35 ¶
    • depends_on proposition 14.73 The extension datum is a $2$-cocycle ¶ ↺
  • depends_on theorem 14.13 Cartan's criterion ¶ ↺
  • proves proof ch:12-lie-groups-fibre-bundles@prooflink-8 ¶
• depends_on theorem A.426 The Kac–Moody cocycle ¶
  • depends_on definition A.424 Loop algebra and residue ¶
    • depends_on definition 14.72 Central extension ¶ ↺
    • depends_on definition 14.10 Killing form ¶ ↺
  • depends_on lemma A.425 The residue of a derivative vanishes ¶
    • depends_on definition A.424 Loop algebra and residue ¶ ↺
    • proves proof app:A-long-proofs@proof-255 ¶
  • depends_on lemma 14.11 Invariance of the Killing form ¶ ↺
  • proves proof app:A-long-proofs@proof-256 ¶
• proves proof app:A-long-proofs@proof-258 ¶

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