theorem 14.16 Invariant tensors give Casimir operators
open in the book ·
parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:481
· p. 547
Rests on
-
depends_on
definition 14.8
Casimir element
¶
-
depends_on
definition 14.5
Universal enveloping algebra
¶
-
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 ¶
-
step_from
equation 14.4
eq:lie-gengrlie
¶
-
depends_on
equation 14.5
eq:lie-structconst
¶
-
depends_on
definition 14.5
Universal enveloping algebra
¶
-
depends_on
definition 14.14
Invariant symmetric tensor
¶
- depends_on equation 14.5 eq:lie-structconst ¶ ↺
- depends_on definition 14.5 Universal enveloping algebra ¶ ↺
- proves proof ch:12-lie-groups-fibre-bundles@proof-4 ¶
Supports
- 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 14.44
Existence: every allowed $j$ is realized
¶
-
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
lemma 14.41
Ladder algebra
¶
- 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 |
→ | Casimir element | declared | parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:490 |
depends_on |
→ | Invariant symmetric tensor | declared | parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:490 |
depends_on |
→ | Universal enveloping algebra | declared | parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:490 |
depends_on |
← | The Casimir elements are not accidentally zero | declared | appendices/A-long-proofs.tex:16081 |
depends_on |
← | The quadratic Casimir | declared | parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:529 |
depends_on |
← | The two Casimir operators of $\mathfrak{su}(3)$ | declared | parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2182 |
proves |
← | ch:12-lie-groups-fibre-bundles@proof-4 | declared | parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:493 |