proposition 14.77 Semisimple algebras admit no nontrivial extension

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3488 · p. 581

Rests on

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 ¶
  - 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 ¶
- proves proof ch:12-lie-groups-fibre-bundles@prooflink-2 ¶
proves proof ch:12-lie-groups-fibre-bundles@prooflink-8 ¶

Supports

depends_on proposition A.428 Uniqueness in degree zero ¶
depends_on proposition 14.84 Properties of a contraction ¶

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 classifying group	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3496
`depends_on`	→	Cartan's criterion	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3496
`depends_on`	←	Uniqueness in degree zero	declared	appendices/A-long-proofs.tex:21039
`depends_on`	←	Properties of a contraction	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3659
`proves`	←	ch:12-lie-groups-fibre-bundles@prooflink-8	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3503