theorem 15.51 Factorisation of invariant forms
open in the book ·
parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1960
· p. 622
Rests on
- depends_on definition 15.3 Tensor product Lie algebra ¶
- proves proof ch:13-lie-algebra-expansions@proof-30 ¶
Supports
- depends_on corollary 15.70 Signature of the factorised form ¶
-
depends_on
example 15.57
$k=2$ recovers the Killing form
¶
- depends_on proposition 15.55 Completeness fails for $k\ge3$ ¶
- depends_on proposition 15.55 Completeness fails for $k\ge3$ ¶ ↺
- depends_on theorem A.64 Completeness of the factorised bilinear forms ¶
-
depends_on
theorem 15.53
The factorised forms are exactly the balanced ones
¶
- depends_on proposition 15.55 Completeness fails for $k\ge3$ ¶ ↺
-
depends_on
theorem 15.58
Nondegeneracy on a resonant subalgebra
¶
- depends_on corollary 15.69 No positive pairing survives the radical ¶
- depends_on remark 15.59 The central charge is what makes the form work ¶
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 |
→ | Tensor product Lie algebra | declared | parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1975 |
depends_on |
← | Signature of the factorised form | declared | parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2617 |
depends_on |
← | $k=2$ recovers the Killing form | declared | parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2188 |
depends_on |
← | Completeness fails for $k\ge3$ | declared | parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2103 |
depends_on |
← | Completeness of the factorised bilinear forms | declared | appendices/A-long-proofs.tex:4215 |
depends_on |
← | The factorised forms are exactly the balanced ones | declared | parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2029 |
depends_on |
← | Nondegeneracy on a resonant subalgebra | declared | parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2200 |
proves |
← | ch:13-lie-algebra-expansions@proof-30 | declared | parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1978 |