theorem A.568 The radical is the isotropy orbit

open in the book · appendices/A-long-proofs.tex:27282 · p. 3065

Rests on

depends_on lemma A.567 The differential of the momentum map along the orbit ¶
- depends_on definition A.561 Coadjoint action and equivariance ¶
  - depends_on definition 4.21 Group ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
  - depends_on definition 14.2 Lie group ¶
  - depends_on definition 24.43 Momentum map ¶
    - depends_on definition 14.2 Lie group ¶ ↺
    - depends_on definition 24.14 Hamiltonian vector field ¶
      - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on equation 22.4 eq:ham-pdot ¶
      - depends_on equation 22.3 eq:ham-qdot ¶
    - depends_on definition 24.17 Symplectomorphism ¶
      - depends_on definition A.70 Pullback ¶
      - depends_on definition 24.8 Symplectic manifold ¶ ↺
- depends_on proposition A.565 Freeness makes every value regular ¶
  - depends_on lemma A.564 Dimension and double orthogonal ¶
    - depends_on definition 24.2 Symplectic vector space ¶
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
    - depends_on theorem 5.40 Rank–nullity ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on lemma 5.38 Exchange and completion ¶
      - proves proof ch:03-linear-algebra-representations@proof-13 ¶
    - proves proof app:A-long-proofs@proof-339 ¶
  - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
    - depends_on definition 24.8 Symplectic manifold ¶ ↺
    - depends_on definition 24.43 Momentum map ¶ ↺
    - depends_on theorem 13.59 Regular value theorem ¶
      - depends_on definition 13.52 Differential; pushforward ¶
      - depends_on definition 13.58 Hypersurface ¶
      - depends_on definition 13.49 Smooth map between manifolds ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-10 ¶
    - proves proof ch:07-symplectic-geometry@proof-15 ¶
  - depends_on theorem 13.59 Regular value theorem ¶ ↺
  - proves proof app:A-long-proofs@proof-340 ¶
- depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
  - depends_on definition 13.124 Integral curve; complete vector field ¶
    - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.82 Vector field ¶
      - depends_on definition 13.81 Vector on a manifold ¶
  - depends_on theorem A.74 Flow of a time-dependent vector field ¶
    - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
      - depends_on definition 9.2 Linear equation; homogeneity ¶
      - depends_on remark 9.3 Normal form and first-order systems ¶
      - depends_on theorem 9.8 Picard–Lindelöf ¶
      - proves proof ch:07-odes-sturm-liouville@proof-3 ¶
    - depends_on lemma A.72 Iterated integral inequality ¶
      - depends_on definition 7.53 $\ee$ ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof app:A-long-proofs@proof-52 ¶
    - depends_on theorem 9.8 Picard–Lindelöf ¶ ↺
    - proves proof app:A-long-proofs@proof-54 ¶
  - depends_on theorem 9.8 Picard–Lindelöf ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-27 ¶
- proves proof app:A-long-proofs@proof-342 ¶
depends_on lemma A.564 Dimension and double orthogonal ¶ ↺
depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶ ↺
proves proof app:A-long-proofs@proof-343 ¶

Supports

depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
- depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The differential of the momentum map along the orbit	declared	appendices/A-long-proofs.tex:27291
`depends_on`	→	Dimension and double orthogonal	declared	appendices/A-long-proofs.tex:27291
`depends_on`	→	The level set of the momentum map is the symplectic orthogonal of the orbit	declared	appendices/A-long-proofs.tex:27291
`depends_on`	←	Existence and uniqueness of the reduced form	declared	appendices/A-long-proofs.tex:27353
`proves`	←	app:A-long-proofs@proof-343	declared	appendices/A-long-proofs.tex:27295