theorem A.330 Cartan's criterion for semisimplicity

open in the book · appendices/A-long-proofs.tex:16171 · p. 2952

Rests on

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 ¶

Supports

depends_on corollary A.344 The algebras $\mathfrak{so}(p,q)$ are semisimple ¶
- depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
- depends_on example A.359 The Lorentz algebra ¶
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 theorem A.410 Whitehead's first and second lemmas ¶
      - depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶ ↺
- 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 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.350 Invariant polynomials on $\mathfrak{g}$ ¶
- depends_on theorem A.351 Chevalley restriction theorem, quoted ¶
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 ¶ ↺
depends_on theorem A.346 Racah ¶
- depends_on example A.359 The Lorentz algebra ¶ ↺
- depends_on example A.357 Rank one: $\mathfrak{su}(2)$ ¶
- depends_on example A.358 Rank two: $\mathfrak{su}(3)$ ¶

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`	→	Killing form	declared	appendices/A-long-proofs.tex:16182
`depends_on`	→	Invariance of the Killing form	declared	appendices/A-long-proofs.tex:16182
`depends_on`	←	The algebras $\mathfrak{so}(p,q)$ are semisimple	declared	appendices/A-long-proofs.tex:16768
`depends_on`	←	Orthogonal splitting of ideals	declared	appendices/A-long-proofs.tex:16693
`depends_on`	←	Invariant polynomials on $\mathfrak{g}$	declared	appendices/A-long-proofs.tex:17056
`depends_on`	←	The trace form is nondegenerate for a faithful representation	declared	appendices/A-long-proofs.tex:19602
`depends_on`	←	Racah	declared	appendices/A-long-proofs.tex:16923
`proves`	←	app:A-long-proofs@proof-209	declared	appendices/A-long-proofs.tex:16617