definition 14.14 Invariant symmetric tensor

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:458 · p. 546

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 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 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 proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶ ↺

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`	→	eq:lie-structconst	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:466
`depends_on`	←	Invariant polynomials on $\mathfrak{g}$	declared	appendices/A-long-proofs.tex:17056
`depends_on`	←	Invariant tensors give Casimir operators	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:490