remark 15.63 The inclusion is strict, and the strictness is physical

Rests on

• depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶
  • depends_on proposition 15.31 Radius and cosmological constant ¶
    • depends_on definition 13.153 Contractions ¶
      • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
        • depends_on definition 13.48 Differentiable manifold ¶
        • depends_on definition 13.84 Tensor ¶
      • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
        • depends_on definition 13.145 Affine connection ¶
        • depends_on definition 13.147 Torsion ¶
        • depends_on proposition 7.105 Clairaut–Schwarz ¶
        • proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
    • depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
      • depends_on definition 13.153 Contractions ¶ ↺
      • depends_on proposition 13.140 Killing's equation ¶
        • depends_on definition 13.139 Killing vector ¶
        • depends_on definition 13.149 Metric compatibility ¶
        • depends_on proposition 13.127 Component formulas ¶
        • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
        • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
        • proves proof ch:11-manifolds-tensors-curvature@proof-37 ¶
      • depends_on proposition 13.154 Symmetries of the curvature ¶
        • depends_on definition 13.149 Metric compatibility ¶ ↺
        • depends_on proposition 13.157 Bianchi identities ¶
        • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
        • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
        • proves proof ch:11-manifolds-tensors-curvature@proof-43 ¶
      • proves proof ch:11-manifolds-tensors-curvature@proof-48 ¶
    • proves proof ch:13-lie-algebra-expansions@proof-18 ¶
  • 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 ¶
        • 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 ¶
  • depends_on theorem 15.29 The three constant-curvature kinematics, in every dimension ¶
    • 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 ¶ ↺
    • proves proof ch:13-lie-algebra-expansions@proof-17 ¶
  • proves proof ch:13-lie-algebra-expansions@proof-19 ¶
• depends_on proposition 15.62 The complex line is not a semigroup algebra ¶
  • depends_on definition 4.18 Semigroup ¶
    • depends_on definition 4.6 Associativity ¶
      • depends_on definition 4.3 Algebraic structure ¶
        • depends_on definition 4.2 Binary operation ¶
    • depends_on definition 4.4 Internal binary operation; magma ¶
      • depends_on definition 4.3 Algebraic structure ¶ ↺
      • depends_on definition 4.2 Binary operation ¶ ↺
  • depends_on proposition 15.60 Semigroup expansion ¶
    • depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
    • depends_on theorem 15.4 The construction produces a Lie algebra ¶
      • depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
      • proves proof ch:13-lie-algebra-expansions@proof-1 ¶
    • proves proof ch:13-lie-algebra-expansions@proof-34 ¶
  • proves proof ch:13-lie-algebra-expansions@proof-35 ¶
• depends_on proposition 15.28 The quadratic family ¶ ↺

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