proposition 15.28 The quadratic family

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:855 · p. 610

Rests on

depends_on corollary 15.11 Semisimplicity criterion ¶
- depends_on lemma 15.12 Artin decomposition ¶
  - depends_on notation 15.1 Standing conventions of this chapter ¶
  - depends_on theorem 8.19 Fundamental theorem of algebra ¶
    - depends_on theorem 8.18 Liouville ¶
      - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - proves proof ch:06-complex-analysis@proof-12 ¶
    - proves proof ch:06-complex-analysis@proof-13 ¶
  - proves proof ch:13-lie-algebra-expansions@proof-7 ¶
- depends_on theorem 15.10 Factorisation of the Killing form ¶
  - 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 ¶
- proves proof ch:13-lie-algebra-expansions@proof-6 ¶
depends_on definition 15.9 Trace form of an algebra ¶ ↺
proves proof ch:13-lie-algebra-expansions@proof-16 ¶

Supports

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 $D=4$ ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Semisimplicity criterion	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:871
`depends_on`	→	Trace form of an algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:871
`depends_on`	←	The expansion parameter is the cosmological constant	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1071
`depends_on`	←	The inclusion is strict, and the strictness is physical	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2358
`depends_on`	←	The three constant-curvature kinematics, in every dimension	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:920
`proves`	←	ch:13-lie-algebra-expansions@proof-16	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:874