equation 14.5 eq:lie-structconst

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:160

connected by a declared semantic edge

Rests on

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 definition 14.14 Invariant symmetric tensor ¶
- depends_on lemma A.350 Invariant polynomials on $\mathfrak{g}$ ¶
  - depends_on theorem A.351 Chevalley restriction theorem, quoted ¶
- depends_on theorem 14.16 Invariant tensors give Casimir operators ¶
  - depends_on corollary A.328 The Casimir elements are not accidentally zero ¶
  - 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 theorem 14.43 The irreducible representations of $\mathfrak{su}(2)$ ¶
    - 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 proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶ ↺
depends_on definition 14.10 Killing form ¶
- 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 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 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 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 theorem 102.12 Which quark combinations can be colour singlets ¶
    - 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 ¶ ↺
depends_on definition 14.5 Universal enveloping algebra ¶
- depends_on definition A.320 Filtration by degree ¶
  - depends_on definition A.325 Symmetric algebra and symbol ¶
    - depends_on corollary A.328 The Casimir elements are not accidentally zero ¶ ↺
    - depends_on definition A.326 Symmetrization ¶
      - depends_on proposition A.327 Symmetrization is an isomorphism of $\mathfrak{g}$-modules ¶
    - depends_on lemma A.349 Lifting generators through the filtration ¶
    - depends_on proposition A.327 Symmetrization is an isomorphism of $\mathfrak{g}$-modules ¶ ↺
    - depends_on proposition A.348 Symmetrization identifies the invariants ¶
      - depends_on lemma A.349 Lifting generators through the filtration ¶ ↺
  - depends_on lemma A.321 The ordered monomials span ¶
- depends_on definition A.326 Symmetrization ¶ ↺
- depends_on definition 14.8 Casimir element ¶
  - depends_on corollary 14.18 A Casimir acts as a number on an irreducible representation ¶
    - depends_on proposition 14.55 The fundamental, the antifundamental and the adjoint ¶ ↺
    - depends_on theorem A.354 Harish-Chandra, quoted: the labels separate ¶
  - depends_on lemma A.347 Central is the same as invariant ¶
    - depends_on proposition A.348 Symmetrization identifies the invariants ¶ ↺
  - depends_on lemma 14.87 A rescaled Casimir stays central ¶
  - depends_on proposition 14.60 The quadratic invariant ¶
    - depends_on theorem 14.65 The two Casimir operators of the Poincaré algebra in $3+1$ dimensions ¶
      - depends_on example A.394 $D = 4$ ¶
  - depends_on theorem A.346 Racah ¶ ↺
  - depends_on theorem A.354 Harish-Chandra, quoted: the labels separate ¶ ↺
  - depends_on theorem 14.16 Invariant tensors give Casimir operators ¶ ↺
  - depends_on theorem 14.21 Racah ¶
    - depends_on proposition 14.22 The rank of $\mathfrak{so}(p,q)$ ¶
      - depends_on lemma 14.68 The algebra fixing a timelike momentum ¶
    - depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶ ↺
    - depends_on theorem A.385 Labels of a massive representation ¶
      - depends_on example A.394 $D = 4$ ¶ ↺
- depends_on lemma A.347 Central is the same as invariant ¶ ↺
- depends_on theorem A.319 Poincaré–Birkhoff–Witt ¶
  - depends_on corollary A.328 The Casimir elements are not accidentally zero ¶ ↺
  - depends_on definition A.325 Symmetric algebra and symbol ¶ ↺
  - depends_on proposition A.327 Symmetrization is an isomorphism of $\mathfrak{g}$-modules ¶ ↺
- depends_on theorem 14.16 Invariant tensors give Casimir operators ¶ ↺
- depends_on theorem 14.7 Poincaré–Birkhoff–Witt ¶
  - depends_on lemma 14.87 A rescaled Casimir stays central ¶ ↺
depends_on lemma 14.102 Two identities for a $\mathfrak{g}$-valued $1$-form ¶
- depends_on theorem 14.107 Bianchi identity ¶
depends_on theorem 14.110 thm:lie-invpoly ¶
- depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶ ↺
- depends_on theorem 14.111 Chern–Weil ¶
  - depends_on definition 14.115 Characteristic forms ¶
  - depends_on definition 14.112 Chern–Simons form ¶
    - depends_on proposition 14.113 The Chern–Simons forms of degree $3$ and $5$ ¶
      - depends_on proposition 14.117 The quadratic invariant and its Chern–Simons form ¶
  - depends_on proposition 14.114 Chern–Weil forms descend to the base ¶
    - depends_on definition 14.115 Characteristic forms ¶ ↺
  - depends_on proposition 106.81 The topological charge is an integer ¶
    - depends_on proposition 106.83 The vacuum is a superposition of winding sectors ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`step_from`	→	eq:lie-gengrlie	declared — demanding that the commutator of two variations be itself a variation of the same form	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:163
`depends_on`	←	Invariant symmetric tensor	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:466
`depends_on`	←	Killing form	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:378
`depends_on`	←	Universal enveloping algebra	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:297
`depends_on`	←	Two identities for a $\mathfrak{g}$-valued $1$-form	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:4230
`depends_on`	←	thm:lie-invpoly	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:4529