definition 14.10 Killing form

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:366 · p. 545

ground object -- no derivation owed

Rests on

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 ¶

Supports

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 corollary 14.12 Total antisymmetry of the structure constants ¶
- depends_on corollary 14.17 The quadratic Casimir ¶
  - depends_on lemma 14.41 Ladder algebra ¶
    - depends_on proposition 14.44 Existence: every allowed $j$ is realized ¶
      - depends_on proposition 14.45 Which representations descend to $\SO(3,\R)$ ¶
    - depends_on theorem 14.43 The irreducible representations of $\mathfrak{su}(2)$ ¶
      - depends_on example A.394 $D = 4$ ¶
      - depends_on example A.357 Rank one: $\mathfrak{su}(2)$ ¶
      - depends_on proposition 14.44 Existence: every allowed $j$ is realized ¶ ↺
      - depends_on theorem 14.47 Clebsch–Gordan series ¶
  - depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶
    - depends_on example A.358 Rank two: $\mathfrak{su}(3)$ ¶
depends_on definition A.424 Loop algebra and residue ¶
- depends_on lemma A.425 The residue of a derivative vanishes ¶
  - depends_on theorem A.426 The Kac–Moody cocycle ¶
    - depends_on proposition A.428 Uniqueness in degree zero ¶
- depends_on theorem A.426 The Kac–Moody cocycle ¶ ↺
depends_on definition A.400 Invariant form, dual basis, Casimir ¶
- depends_on lemma A.401 The invariance identity ¶
  - depends_on theorem A.403 Vanishing when the Casimir is invertible ¶
    - depends_on theorem A.410 Whitehead's first and second lemmas ¶ ↺
depends_on lemma A.350 Invariant polynomials on $\mathfrak{g}$ ¶
- depends_on theorem A.351 Chevalley restriction theorem, quoted ¶
depends_on lemma 14.11 Invariance of the Killing form ¶
- depends_on corollary 14.12 Total antisymmetry of the structure constants ¶ ↺
- depends_on definition A.400 Invariant form, dual basis, Casimir ¶ ↺
- depends_on lemma A.427 Invariant bilinear forms on a simple algebra ¶
  - depends_on proposition A.428 Uniqueness in degree zero ¶ ↺
- depends_on theorem A.330 Cartan's criterion for semisimplicity ¶
  - 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.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.350 Invariant polynomials on $\mathfrak{g}$ ¶ ↺
  - 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: $\mathfrak{su}(3)$ ¶ ↺
- depends_on theorem A.426 The Kac–Moody cocycle ¶ ↺
depends_on proposition A.334 Nondegeneracy implies no abelian ideal ¶
depends_on proposition 14.84 Properties of a contraction ¶
depends_on theorem A.330 Cartan's criterion for semisimplicity ¶ ↺
depends_on theorem 14.13 Cartan's criterion ¶
- depends_on corollary 14.17 The quadratic Casimir ¶ ↺
- depends_on lemma A.427 Invariant bilinear forms on a simple algebra ¶ ↺
- depends_on proposition 14.53 Killing form of $\mathfrak{su}(3)$ ¶
  - depends_on proposition 14.55 The fundamental, the antifundamental and the adjoint ¶
    - depends_on corollary A.377 The adjoint representation is irreducible ¶
    - depends_on example A.358 Rank two: $\mathfrak{su}(3)$ ¶ ↺
    - depends_on theorem 14.57 $\vect{3}\otimes\bar{\vect{3}} = \vect{1}\oplus\vect{8}$ ¶
      - depends_on proposition 102.6 Mesons are $q\bar{q}$, baryons are $qqq$ ¶
    - depends_on theorem 102.12 Which quark combinations can be colour singlets ¶
      - depends_on corollary 102.13 The selection rule of the eightfold way ¶
  - depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶ ↺
- depends_on proposition 14.77 Semisimple algebras admit no nontrivial extension ¶
  - depends_on proposition A.428 Uniqueness in degree zero ¶ ↺
  - depends_on proposition 14.84 Properties of a contraction ¶ ↺

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`	→	eq:lie-structconst	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:378
`depends_on`	←	The Killing form of a simple ideal	declared	appendices/A-long-proofs.tex:16739
`depends_on`	←	Total antisymmetry of the structure constants	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:420
`depends_on`	←	Loop algebra and residue	declared	appendices/A-long-proofs.tex:20878
`depends_on`	←	Invariant form, dual basis, Casimir	declared	appendices/A-long-proofs.tex:19547
`depends_on`	←	Invariant polynomials on $\mathfrak{g}$	declared	appendices/A-long-proofs.tex:17056
`depends_on`	←	Invariance of the Killing form	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:387
`depends_on`	←	Nondegeneracy implies no abelian ideal	declared	appendices/A-long-proofs.tex:16278
`depends_on`	←	Properties of a contraction	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3659
`depends_on`	←	Cartan's criterion for semisimplicity	declared	appendices/A-long-proofs.tex:16182
`depends_on`	←	Cartan's criterion	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:442