proposition 24.51 Prequantization is a Lie-algebra homomorphism

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1716 · p. 868

Rests on

depends_on definition 24.50 Prequantum operator ¶
- depends_on definition 24.14 Hamiltonian vector field ¶
  - depends_on definition 24.8 Symplectic manifold ¶
    - depends_on definition 13.105 Closed form ¶
      - depends_on definition 13.103 Exterior derivative ¶
      - depends_on definition 13.98 $k$-form ¶
    - depends_on definition 13.98 $k$-form ¶ ↺
    - depends_on definition 24.2 Symplectic vector space ¶
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
  - depends_on equation 22.4 eq:ham-pdot ¶
  - depends_on equation 22.3 eq:ham-qdot ¶
- depends_on definition 24.49 Prequantum datum ¶
  - depends_on definition 14.103 Connection $1$-form ¶
    - depends_on definition 14.97 Principal bundle ¶
      - depends_on definition 14.95 Fibre bundle ¶
    - depends_on notation 14.101 Algebra-valued forms and their bracket ¶
  - depends_on definition 14.105 Curvature $2$-form ¶
    - depends_on definition 14.103 Connection $1$-form ¶ ↺
    - depends_on notation 14.101 Algebra-valued forms and their bracket ¶ ↺
  - depends_on definition 24.8 Symplectic manifold ¶ ↺
depends_on equation 24.8 eq:sym-bracket-homomorphism ¶
depends_on equation 24.34 eq:sym-prequantum-condition ¶
proves proof ch:07-symplectic-geometry@proof-16 ¶

Supports

depends_on example 24.52 The prequantum line and the Schrödinger representation ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Prequantum operator	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1728
`depends_on`	→	eq:sym-bracket-homomorphism	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1728
`depends_on`	→	eq:sym-prequantum-condition	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1728
`depends_on`	←	The prequantum line and the Schrödinger representation	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1803
`proves`	←	ch:07-symplectic-geometry@proof-16	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1732