lemma A.427 Invariant bilinear forms on a simple algebra

open in the book · appendices/A-long-proofs.tex:20993 · p. 3001

Rests on

depends_on lemma 14.11 Invariance of the Killing form ¶
- 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@proof-2 ¶
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 ¶
depends_on theorem 5.158 Schur's first lemma ¶
- depends_on definition 5.37 Linear transformation ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 5.147 Invariant subspace ¶
  - depends_on definition 5.7 Vector subspace ¶
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on definition 5.141 Representation of a group ¶
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on definition 4.43 Group homomorphism ¶
      - depends_on definition 4.21 Group ¶ ↺
      - depends_on definition 3.43 Map ¶
    - depends_on definition 4.48 Symmetric group ¶
      - depends_on definition 3.47 Bijective map ¶
- depends_on definition 5.148 Irreducible representation ¶
  - depends_on definition 5.147 Invariant subspace ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-67 ¶
proves proof app:A-long-proofs@proof-257 ¶

Supports

depends_on proposition A.428 Uniqueness in degree zero ¶

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`	→	Invariance of the Killing form	declared	appendices/A-long-proofs.tex:21002
`depends_on`	→	Cartan's criterion	declared	appendices/A-long-proofs.tex:21002
`depends_on`	→	Schur's first lemma	declared	appendices/A-long-proofs.tex:21002
`depends_on`	←	Uniqueness in degree zero	declared	appendices/A-long-proofs.tex:21039
`proves`	←	app:A-long-proofs@proof-257	declared	appendices/A-long-proofs.tex:21005