lemma A.402 The trace form is nondegenerate for a faithful representation

open in the book · appendices/A-long-proofs.tex:19593 · p. 2987

Rests on

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 ¶
    - step_from equation 14.4 eq:lie-gengrlie ¶
      - step_from equation 14.3 eq:lie-generators ¶
      - step_from equation 14.2 eq:lie-varincor ¶
- 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 ¶
      - depends_on definition 5.37 Linear transformation ¶
      - proves proof app:A-long-proofs@proof-203 ¶
    - proves proof app:A-long-proofs@proof-204 ¶
  - 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 ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
    - proves proof app:A-long-proofs@proof-205 ¶
  - proves proof app:A-long-proofs@proof-206 ¶
- proves proof app:A-long-proofs@proof-208 ¶
proves proof app:A-long-proofs@proof-241 ¶

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 ¶

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`	→	Cartan's criterion for semisimplicity	declared	appendices/A-long-proofs.tex:19602
`depends_on`	→	Cartan's criterion for solvability	declared	appendices/A-long-proofs.tex:19602
`depends_on`	←	Splitting a submodule of codimension one	declared	appendices/A-long-proofs.tex:19773
`proves`	←	app:A-long-proofs@proof-241	declared	appendices/A-long-proofs.tex:19605