proposition 14.84 Properties of a contraction

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3629 · p. 583

Rests on

depends_on definition 14.83 Contraction ¶
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 ¶
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 ¶
  - depends_on definition 14.10 Killing form ¶ ↺
  - proves proof ch:12-lie-groups-fibre-bundles@prooflink-2 ¶
- proves proof ch:12-lie-groups-fibre-bundles@prooflink-8 ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-37 ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Contraction	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3659
`depends_on`	→	Killing form	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3659
`depends_on`	→	Semisimple algebras admit no nontrivial extension	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3659
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-37	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3662