lemma A.337 A nilpotent map has nilpotent adjoint

open in the book · appendices/A-long-proofs.tex:16405 · p. 2954

Rests on

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 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
  - depends_on definition 4.32 Field ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.18 Semigroup ¶ ↺
  - depends_on definition 4.31 Module ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.28 Distributivity ¶ ↺
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on definition 4.30 Ring ¶ ↺
proves proof app:A-long-proofs@proof-205 ¶

Supports

depends_on theorem A.338 Engel ¶
- depends_on theorem A.340 Cartan's criterion for solvability ¶
  - depends_on corollary A.342 Orthogonal splitting of ideals ¶
    - depends_on corollary A.343 The Killing form of a simple ideal ¶
      - depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶
    - depends_on lemma A.406 Splitting a submodule of codimension one ¶
      - depends_on theorem A.407 Weyl's complete reducibility theorem ¶
    - depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶ ↺
    - depends_on proposition A.408 The two groups vanish for the trivial module ¶
    - depends_on theorem A.407 Weyl's complete reducibility theorem ¶ ↺
  - depends_on lemma A.402 The trace form is nondegenerate for a faithful representation ¶
    - depends_on lemma A.406 Splitting a submodule of codimension one ¶ ↺

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`	→	Linear transformation	declared	appendices/A-long-proofs.tex:16409
`depends_on`	←	Engel	declared	appendices/A-long-proofs.tex:16427
`proves`	←	app:A-long-proofs@proof-205	declared	appendices/A-long-proofs.tex:16412