example 15.64 The whole chapter at $D=4$

Rests on

• depends_on corollary 15.36 Bargmann in $3{+}1$, extended Bargmann in $2{+}1$ ¶
  • depends_on corollary 15.35 Where the central charges are ¶
    • 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 theorem 15.34 The truncation tower is a tower of central extensions ¶
      • depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶
      • depends_on definition 15.20 Resonant subalgebra ¶
        • depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶ ↺
        • depends_on definition 15.14 Truncated polynomial algebra and Takiff algebra ¶
      • depends_on definition 14.72 Central extension ¶
      • depends_on proposition 15.6 Functoriality ¶
        • depends_on definition 15.3 Tensor product Lie algebra ¶
        • depends_on theorem 15.4 The construction produces a Lie algebra ¶
        • proves proof ch:13-lie-algebra-expansions@proof-2 ¶
      • depends_on proposition 14.73 The extension datum is a $2$-cocycle ¶
        • depends_on definition 14.72 Central extension ¶ ↺
        • proves proof ch:12-lie-groups-fibre-bundles@proof-35 ¶
      • proves proof ch:13-lie-algebra-expansions@proof-20 ¶
    • proves proof ch:13-lie-algebra-expansions@proof-21 ¶
  • depends_on corollary 15.27 The flat limit ¶
    • depends_on definition 15.24 The four kinematical gradings ¶ ↺
    • depends_on equation 14.99 eq:lie-poincare ¶
    • 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 15.26 The contraction along a grading ¶
      • depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶ ↺
      • depends_on definition 15.20 Resonant subalgebra ¶ ↺
      • depends_on equation 14.125 eq:lie-contraction ¶
      • depends_on proposition 15.15 Takiff brackets ¶
        • depends_on definition 15.14 Truncated polynomial algebra and Takiff algebra ¶ ↺
        • depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
        • depends_on proposition 15.8 The radical of $A$ is solvable in $\mathfrak{g}_{A}$ ¶
        • proves proof ch:13-lie-algebra-expansions@proof-9 ¶
      • proves proof ch:13-lie-algebra-expansions@proof-14 ¶
    • proves proof ch:13-lie-algebra-expansions@proof-15 ¶
  • depends_on proposition 25.14 The Galilei cocycle is not a coboundary ¶
    • depends_on definition 14.72 Central extension ¶ ↺
    • depends_on equation 25.15 eq:pq-galilei-central ¶
    • depends_on proposition 14.73 The extension datum is a $2$-cocycle ¶ ↺
    • proves proof ch:08-poisson-quantum-bridge@proof-7 ¶
  • depends_on theorem 15.34 The truncation tower is a tower of central extensions ¶ ↺
  • proves proof ch:13-lie-algebra-expansions@proof-22 ¶
• depends_on corollary 15.35 Where the central charges are ¶ ↺
• 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 definition 15.3 Tensor product Lie algebra ¶ ↺
    • depends_on proposition 15.23 Brackets in the kinematical splitting ¶ ↺
    • 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 15.31 Radius and cosmological constant ¶ ↺
• depends_on theorem 15.29 The three constant-curvature kinematics, in every dimension ¶ ↺

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