proposition A.408 The two groups vanish for the trivial module

open in the book · appendices/A-long-proofs.tex:19896 · p. 2990

Rests on

depends_on corollary A.342 Orthogonal splitting of ideals ¶
- depends_on theorem A.330 Cartan's criterion for semisimplicity ¶
  - 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 app:A-long-proofs@proof-209 ¶
- depends_on theorem A.340 Cartan's criterion for solvability ¶
  - depends_on lemma A.339 Trace lemma ¶
    - depends_on lemma A.336 The adjoint of a Jordan decomposition ¶
      - depends_on theorem A.335 Jordan decomposition ¶
      - proves proof app:A-long-proofs@proof-204 ¶
    - depends_on theorem A.335 Jordan decomposition ¶ ↺
    - proves proof app:A-long-proofs@proof-207 ¶
  - depends_on theorem A.338 Engel ¶
    - depends_on definition A.331 Derived series, solvable, ideal ¶
    - depends_on lemma A.337 A nilpotent map has nilpotent adjoint ¶
      - depends_on definition 5.37 Linear transformation ¶
      - proves proof app:A-long-proofs@proof-205 ¶
    - proves proof app:A-long-proofs@proof-206 ¶
  - proves proof app:A-long-proofs@proof-208 ¶
- proves proof app:A-long-proofs@proof-210 ¶
depends_on proposition 14.76 $H^{2}$ classifies the central extensions ¶
- depends_on definition 14.72 Central extension ¶
- 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 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 proposition 14.73 The extension datum is a $2$-cocycle ¶ ↺
- proves proof ch:12-lie-groups-fibre-bundles@proof-36 ¶
depends_on theorem A.407 Weyl's complete reducibility theorem ¶
- depends_on corollary A.342 Orthogonal splitting of ideals ¶ ↺
- depends_on lemma A.406 Splitting a submodule of codimension one ¶
  - depends_on corollary A.342 Orthogonal splitting of ideals ¶ ↺
  - depends_on lemma A.405 Schur, algebra form ¶
    - depends_on theorem 5.158 Schur's first lemma ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.147 Invariant subspace ¶
      - depends_on definition 5.148 Irreducible representation ¶
      - proves proof ch:03-linear-algebra-representations@proof-67 ¶
    - proves proof app:A-long-proofs@proof-243 ¶
  - depends_on lemma A.402 The trace form is nondegenerate for a faithful representation ¶
    - depends_on theorem A.330 Cartan's criterion for semisimplicity ¶ ↺
    - depends_on theorem A.340 Cartan's criterion for solvability ¶ ↺
    - proves proof app:A-long-proofs@proof-241 ¶
  - proves proof app:A-long-proofs@proof-244 ¶
- proves proof app:A-long-proofs@proof-245 ¶
proves proof app:A-long-proofs@proof-246 ¶

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type	direction	node	provenance	where
`depends_on`	→	Orthogonal splitting of ideals	declared	appendices/A-long-proofs.tex:19902
`depends_on`	→	$H^{2}$ classifies the central extensions	declared	appendices/A-long-proofs.tex:19902
`depends_on`	→	Weyl's complete reducibility theorem	declared	appendices/A-long-proofs.tex:19902
`proves`	←	app:A-long-proofs@proof-246	declared	appendices/A-long-proofs.tex:19906