theorem 15.29 The three constant-curvature kinematics, in every dimension

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:899 · p. 610

Rests on

depends_on definition 15.24 The four kinematical gradings ¶
- depends_on notation 15.22 Generators of $\mathfrak{so}(D-1,2)$ ¶
  - depends_on equation 14.90 eq:lie-algebrasopq ¶
depends_on definition 15.3 Tensor product Lie algebra ¶
depends_on proposition 15.23 Brackets in the kinematical splitting ¶
- depends_on equation 14.90 eq:lie-algebrasopq ¶ ↺
- depends_on equation 14.128 eq:lie-kinematical-constants ¶
- depends_on notation 15.22 Generators of $\mathfrak{so}(D-1,2)$ ¶ ↺
- proves proof ch:13-lie-algebra-expansions@proof-12 ¶
depends_on proposition 15.28 The quadratic family ¶
- 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 ¶
      - 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 ¶
proves proof ch:13-lie-algebra-expansions@proof-17 ¶

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 example 15.64 The whole chapter at $D=4$ ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	The four kinematical gradings	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:920
`depends_on`	→	Tensor product Lie algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:920
`depends_on`	→	Brackets in the kinematical splitting	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:920
`depends_on`	→	The quadratic family	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:920
`depends_on`	←	The expansion parameter is the cosmological constant	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1071
`depends_on`	←	The whole chapter at $D=4$	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2407
`proves`	←	ch:13-lie-algebra-expansions@proof-17	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:923