definition 15.3 Tensor product Lie algebra

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Supports

• depends_on definition 15.52 Balanced form ¶
  • depends_on theorem 15.53 The factorised forms are exactly the balanced ones ¶
    • depends_on proposition 15.55 Completeness fails for $k\ge3$ ¶
• depends_on definition 15.14 Truncated polynomial algebra and Takiff algebra ¶
  • depends_on corollary 15.69 No positive pairing survives the radical ¶
  • depends_on definition 15.20 Resonant subalgebra ¶
    • depends_on example 15.61 Maurer–Cartan expansion ¶
    • depends_on proposition 15.26 The contraction along a grading ¶
      • 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 corollary 15.27 The flat limit ¶
        • depends_on corollary 15.36 Bargmann in $3{+}1$, extended Bargmann in $2{+}1$ ¶
    • depends_on proposition 15.21 Resonance ¶
    • depends_on theorem 15.58 Nondegeneracy on a resonant subalgebra ¶
      • depends_on corollary 15.69 No positive pairing survives the radical ¶ ↺
      • depends_on remark 15.59 The central charge is what makes the form work ¶
    • depends_on theorem 15.34 The truncation tower is a tower of central extensions ¶
      • 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 corollary 15.36 Bargmann in $3{+}1$, extended Bargmann in $2{+}1$ ¶ ↺
        • depends_on example 15.64 The whole chapter at $D=4$ ¶
  • depends_on example 15.61 Maurer–Cartan expansion ¶ ↺
  • depends_on proposition 15.15 Takiff brackets ¶
    • depends_on example 15.16 The Maxwell tower ¶
    • depends_on proposition 15.26 The contraction along a grading ¶ ↺
    • depends_on proposition 15.21 Resonance ¶ ↺
• depends_on example 15.61 Maurer–Cartan expansion ¶ ↺
• depends_on proposition 15.6 Functoriality ¶
  • depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶
  • depends_on theorem 15.34 The truncation tower is a tower of central extensions ¶ ↺
• depends_on proposition 15.41 Resonance, general form ¶
  • 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 proposition 15.7 Ideals of $A$ become ideals of $\mathfrak{g}_{A}$ ¶
  • depends_on proposition 15.8 The radical of $A$ is solvable in $\mathfrak{g}_{A}$ ¶
    • depends_on proposition 15.15 Takiff brackets ¶ ↺
    • depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶ ↺
• depends_on proposition 15.8 The radical of $A$ is solvable in $\mathfrak{g}_{A}$ ¶ ↺
• depends_on proposition 15.60 Semigroup expansion ¶
  • depends_on proposition 15.62 The complex line is not a semigroup algebra ¶
    • depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶
• depends_on proposition 15.15 Takiff brackets ¶ ↺
• depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶ ↺
• depends_on theorem 15.54 Completeness in the bilinear case ¶
  • depends_on definition 15.66 Gram form of a functional ¶
    • depends_on corollary 15.70 Signature of the factorised form ¶
    • depends_on lemma 15.67 Multiplication is $B_{\psi}$-self-adjoint ¶
      • depends_on theorem 15.68 Positive Gram forms are positive sums of real characters ¶
        • depends_on corollary 15.69 No positive pairing survives the radical ¶ ↺
    • depends_on theorem 15.68 Positive Gram forms are positive sums of real characters ¶ ↺
• 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 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 $D=4$ ¶ ↺
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