definition 14.5 Universal enveloping algebra

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:281 · p. 544

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 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 proposition A.348 Symmetrization identifies the invariants ¶
  - 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.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 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 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 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.354 Harish-Chandra, quoted: the labels separate ¶ ↺
- 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 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 definition A.384 The mass shell and the standard momentum ¶
      - depends_on lemma A.386 The little group ¶
      - depends_on theorem A.385 Labels of a massive representation ¶
  - 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 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 ¶ ↺

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:297
`depends_on`	←	Filtration by degree	declared	appendices/A-long-proofs.tex:15678
`depends_on`	←	Symmetrization	declared	appendices/A-long-proofs.tex:16005
`depends_on`	←	Casimir element	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:339
`depends_on`	←	Central is the same as invariant	declared	appendices/A-long-proofs.tex:16937
`depends_on`	←	Poincaré–Birkhoff–Witt	declared	appendices/A-long-proofs.tex:15662
`depends_on`	←	Invariant tensors give Casimir operators	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:490
`depends_on`	←	Poincaré–Birkhoff–Witt	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:319