lemma A.405 Schur, algebra form

open in the book · appendices/A-long-proofs.tex:19747 · p. 2989

Rests on

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-243 ¶

Supports

depends_on lemma A.406 Splitting a submodule of codimension one ¶
- depends_on theorem A.407 Weyl's complete reducibility theorem ¶
  - depends_on proposition A.408 The two groups vanish for the trivial module ¶
  - depends_on theorem A.410 Whitehead's first and second lemmas ¶
    - depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶
- depends_on theorem A.410 Whitehead's first and second lemmas ¶ ↺

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`	→	Schur's first lemma	declared	appendices/A-long-proofs.tex:19753
`depends_on`	←	Splitting a submodule of codimension one	declared	appendices/A-long-proofs.tex:19773
`depends_on`	←	A Casimir invertible on a nontrivial irreducible module	declared	appendices/A-long-proofs.tex:19965
`proves`	←	app:A-long-proofs@proof-243	declared	appendices/A-long-proofs.tex:19756