lemma A.409 A Casimir invertible on a nontrivial irreducible module

open in the book · appendices/A-long-proofs.tex:19959 · p. 2991

Rests on

depends_on corollary A.343 The Killing form of a simple ideal ¶
- depends_on corollary A.342 Orthogonal splitting of ideals ¶
  - depends_on theorem A.330 Cartan's criterion for semisimplicity ¶
    - depends_on definition 14.10 Killing form ¶
      - depends_on equation 14.5 eq:lie-structconst ¶
    - 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-209 ¶
  - depends_on theorem A.340 Cartan's criterion for solvability ¶
    - depends_on lemma A.339 Trace lemma ¶
      - depends_on lemma A.336 The adjoint of a Jordan decomposition ¶
      - depends_on theorem A.335 Jordan decomposition ¶
      - proves proof app:A-long-proofs@proof-207 ¶
    - depends_on theorem A.338 Engel ¶
      - depends_on definition A.331 Derived series, solvable, ideal ¶
      - depends_on lemma A.337 A nilpotent map has nilpotent adjoint ¶
      - proves proof app:A-long-proofs@proof-206 ¶
    - proves proof app:A-long-proofs@proof-208 ¶
  - proves proof app:A-long-proofs@proof-210 ¶
- depends_on definition 14.10 Killing form ¶ ↺
- proves proof app:A-long-proofs@proof-211 ¶
depends_on corollary A.342 Orthogonal splitting of ideals ¶ ↺
depends_on lemma A.405 Schur, algebra form ¶
- 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 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.147 Invariant subspace ¶
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶
    - 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 4.48 Symmetric group ¶
  - 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-243 ¶
proves proof app:A-long-proofs@proof-247 ¶

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`	→	The Killing form of a simple ideal	declared	appendices/A-long-proofs.tex:19965
`depends_on`	→	Orthogonal splitting of ideals	declared	appendices/A-long-proofs.tex:19965
`depends_on`	→	Schur, algebra form	declared	appendices/A-long-proofs.tex:19965
`depends_on`	←	Whitehead's first and second lemmas	declared	appendices/A-long-proofs.tex:20053
`proves`	←	app:A-long-proofs@proof-247	declared	appendices/A-long-proofs.tex:19969