example A.359 The Lorentz algebra

open in the book · appendices/A-long-proofs.tex:17281 · p. 2963

example -- no derivation owed by its kind

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depends_on corollary A.344 The algebras $\mathfrak{so}(p,q)$ are semisimple ¶
- depends_on equation 14.90 eq:lie-algebrasopq ¶
- depends_on proposition 14.59 The Killing fields of a flat pseudo-Euclidean space ¶
  - depends_on definition 13.139 Killing vector ¶
    - depends_on definition 13.120 Isometry ¶
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
    - depends_on equation 13.267 eq:mfd-lie-def ¶
  - depends_on equation 14.86 eq:lie-sopq-gen ¶
  - depends_on proposition 13.140 Killing's equation ¶
    - depends_on definition 13.139 Killing vector ¶ ↺
    - depends_on definition 13.149 Metric compatibility ¶
      - depends_on definition 13.145 Affine connection ¶
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      - depends_on definition 13.146 Parallel transport and autoparallels ¶
    - depends_on proposition 13.127 Component formulas ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.90 Mixed tensor ¶
      - depends_on equation 13.267 eq:mfd-lie-def ¶ ↺
      - proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
    - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      - depends_on definition 13.149 Metric compatibility ¶ ↺
      - depends_on definition 13.147 Torsion ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
    - 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 ¶
    - proves proof ch:11-manifolds-tensors-curvature@proof-37 ¶
  - proves proof ch:12-lie-groups-fibre-bundles@proof-27 ¶
- depends_on theorem A.330 Cartan's criterion for semisimplicity ¶
  - depends_on definition 14.10 Killing form ¶
    - depends_on equation 14.5 eq:lie-structconst ¶
      - step_from equation 14.4 eq:lie-gengrlie ¶
  - depends_on lemma 14.11 Invariance of the Killing form ¶
    - depends_on definition 14.10 Killing form ¶ ↺
    - proves proof ch:12-lie-groups-fibre-bundles@proof-2 ¶
  - proves proof app:A-long-proofs@proof-209 ¶
- proves proof app:A-long-proofs@proof-212 ¶
depends_on equation 14.22 eq:lie-sopq-rank ¶
depends_on theorem A.346 Racah ¶
- depends_on definition 14.8 Casimir element ¶
  - depends_on definition 14.5 Universal enveloping algebra ¶
    - depends_on equation 14.5 eq:lie-structconst ¶ ↺
- depends_on definition 14.20 Cartan subalgebra and rank ¶
- depends_on theorem A.330 Cartan's criterion for semisimplicity ¶ ↺
- proves proof app:A-long-proofs@proof-217 ¶

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type	direction	node	provenance	where
`depends_on`	→	The algebras $\mathfrak{so}(p,q)$ are semisimple	declared	appendices/A-long-proofs.tex:17297
`depends_on`	→	eq:lie-sopq-rank	declared	appendices/A-long-proofs.tex:17297
`depends_on`	→	Racah	declared	appendices/A-long-proofs.tex:17297