theorem A.403 Vanishing when the Casimir is invertible

open in the book · appendices/A-long-proofs.tex:19630 · p. 2988

Rests on

depends_on definition A.397 Cochains and the differential ¶
- 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 lemma A.401 The invariance identity ¶
- depends_on definition A.400 Invariant form, dual basis, Casimir ¶
  - 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-240 ¶
proves proof app:A-long-proofs@proof-242 ¶

Supports

depends_on theorem A.410 Whitehead's first and second lemmas ¶
- depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶

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`	→	Cochains and the differential	declared	appendices/A-long-proofs.tex:19638
`depends_on`	→	The invariance identity	declared	appendices/A-long-proofs.tex:19638
`depends_on`	←	Whitehead's first and second lemmas	declared	appendices/A-long-proofs.tex:20053
`proves`	←	app:A-long-proofs@proof-242	declared	appendices/A-long-proofs.tex:19641