proposition 24.5 Properties of the symplectic group

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:158 · p. 850

Rests on

depends_on definition 24.4 Symplectic group ¶
- depends_on definition 5.135 The symplectic group ¶
  - depends_on definition 5.113 Non-degenerate form ¶
    - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
  - depends_on equation 5.138 eq:lin-bilinear-expansion ¶
  - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
    - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on definition 5.113 Non-degenerate form ¶ ↺
    - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
      - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
    - proves proof ch:03-linear-algebra-representations@proof-54 ¶
- depends_on proposition 24.3 Even dimension and the canonical basis ¶
  - depends_on definition 24.2 Symplectic vector space ¶
    - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
  - depends_on equation 22.56 eq:ham-symplectic-matrix ¶
  - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
  - proves proof ch:07-symplectic-geometry@proof-1 ¶
depends_on proposition 5.139 Dimension of the symplectic group ¶
- depends_on definition 5.127 Lie algebra ¶
  - depends_on definition 5.121 Algebra ¶
    - depends_on definition 4.4 Internal binary operation; magma ¶
      - depends_on definition 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
- depends_on definition 5.135 The symplectic group ¶ ↺
- depends_on proposition 5.125 prop:lin-matrix-algebra ¶
  - depends_on definition 5.121 Algebra ¶ ↺
  - depends_on definition 5.37 Linear transformation ¶ ↺
  - depends_on equation 5.150 eq:lin-ctesesc2 ¶
  - depends_on equation 5.74 eq:lin-matrix-rep ¶
  - depends_on proposition 5.45 prop:lin-matrix-unique ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on equation 5.29 eq:lin-indepli2 ¶
    - depends_on equation 5.74 eq:lin-matrix-rep ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-16 ¶
  - proves proof ch:03-linear-algebra-representations@proof-55 ¶
- proves proof ch:03-linear-algebra-representations@proof-62 ¶
depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶
- depends_on definition 5.135 The symplectic group ¶ ↺
- depends_on lemma 5.137 The alternating top form is unique up to scale ¶
  - depends_on definition 5.15 Basis ¶ ↺
  - depends_on definition 5.110 Bilinear map ¶ ↺
  - depends_on equation 5.19 eq:lin-leibniz-det ¶
  - proves proof ch:03-linear-algebra-representations@proof-60 ¶
- depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-61 ¶
proves proof ch:07-symplectic-geometry@proof-2 ¶

Supports

depends_on remark 24.6 Connectedness, and what it rests on ¶
depends_on remark 24.7 Contrast with the orthogonal group ¶

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`	→	Symplectic group	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:165
`depends_on`	→	Dimension of the symplectic group	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:165
`depends_on`	→	A symplectic transformation has determinant $+1$	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:165
`depends_on`	←	Connectedness, and what it rests on	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:230
`depends_on`	←	Contrast with the orthogonal group	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:240
`proves`	←	ch:07-symplectic-geometry@proof-2	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:169