theorem 15.10 Factorisation of the Killing form

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:252 · p. 604

Rests on

depends_on definition 15.3 Tensor product Lie algebra ¶
depends_on definition 15.9 Trace form of an algebra ¶
proves proof ch:13-lie-algebra-expansions@proof-5 ¶

Supports

depends_on corollary 15.11 Semisimplicity criterion ¶
- depends_on proposition 15.42 The $\Gamma$-graded commutative algebras ¶
  - depends_on corollary 15.46 The sign data, hypothesis (H3), and the count eleven ¶
  - depends_on theorem 15.43 The three constants of the classification are the three structure constants of $A$ ¶
    - depends_on corollary 15.46 The sign data, hypothesis (H3), and the count eleven ¶ ↺
    - depends_on corollary 15.44 The eight vertices are the eight degenerations of $A$ ¶
      - depends_on remark 15.45 A correction this forces: the general splitting gives the static algebra, not Carroll ¶
- depends_on proposition 15.28 The quadratic family ¶
  - depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶
    - depends_on example 15.64 The whole chapter at $D=4$ ¶
    - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶
    - depends_on remark 15.33 Which branch nature is on ¶
  - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶ ↺
  - depends_on theorem 15.29 The three constant-curvature kinematics, in every dimension ¶
    - depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶ ↺
    - depends_on example 15.64 The whole chapter at $D=4$ ¶ ↺
- depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶
depends_on example 15.57 $k=2$ recovers the Killing form ¶
- depends_on proposition 15.55 Completeness fails for $k\ge3$ ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Tensor product Lie algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:261
`depends_on`	→	Trace form of an algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:261
`depends_on`	←	Semisimplicity criterion	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:295
`depends_on`	←	$k=2$ recovers the Killing form	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2188
`proves`	←	ch:13-lie-algebra-expansions@proof-5	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:264