definition A.400 Invariant form, dual basis, Casimir

open in the book · appendices/A-long-proofs.tex:19526 · p. 2987

ground object -- no derivation owed

Rests on

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 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 ¶

Supports

depends_on lemma A.401 The invariance identity ¶
- depends_on theorem A.403 Vanishing when the Casimir is invertible ¶
  - 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`	→	Killing form	declared	appendices/A-long-proofs.tex:19547
`depends_on`	→	Invariance of the Killing form	declared	appendices/A-long-proofs.tex:19547
`depends_on`	←	The invariance identity	declared	appendices/A-long-proofs.tex:19560