theorem A.410 Whitehead's first and second lemmas

open in the book · appendices/A-long-proofs.tex:20044 · p. 2991

Rests on

depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶
- 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 lemma 14.11 Invariance of the Killing form ¶
      - 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 theorem A.338 Engel ¶
      - 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 5.147 Invariant subspace ¶
      - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.141 Representation of a 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 ¶
depends_on theorem A.403 Vanishing when the Casimir is invertible ¶
- 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 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 lemma 14.11 Invariance of the Killing form ¶ ↺
  - proves proof app:A-long-proofs@proof-240 ¶
- proves proof app:A-long-proofs@proof-242 ¶
depends_on theorem A.407 Weyl's complete reducibility theorem ¶
- depends_on corollary A.342 Orthogonal splitting of ideals ¶ ↺
- depends_on lemma A.406 Splitting a submodule of codimension one ¶
  - depends_on corollary A.342 Orthogonal splitting of ideals ¶ ↺
  - depends_on lemma A.405 Schur, algebra form ¶ ↺
  - depends_on lemma A.402 The trace form is nondegenerate for a faithful representation ¶
    - depends_on theorem A.330 Cartan's criterion for semisimplicity ¶ ↺
    - depends_on theorem A.340 Cartan's criterion for solvability ¶ ↺
    - proves proof app:A-long-proofs@proof-241 ¶
  - proves proof app:A-long-proofs@proof-244 ¶
- proves proof app:A-long-proofs@proof-245 ¶
proves proof app:A-long-proofs@proof-248 ¶

Supports

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`	→	A Casimir invertible on a nontrivial irreducible module	declared	appendices/A-long-proofs.tex:20053
`depends_on`	→	Vanishing when the Casimir is invertible	declared	appendices/A-long-proofs.tex:20053
`depends_on`	→	Weyl's complete reducibility theorem	declared	appendices/A-long-proofs.tex:20053
`depends_on`	←	Semisimple algebras admit no nontrivial extension	declared	appendices/A-long-proofs.tex:20107
`proves`	←	app:A-long-proofs@proof-248	declared	appendices/A-long-proofs.tex:20057