definition 5.127 Lie algebra

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:5445 · p. 160

ground object -- no derivation owed

Rests on

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 4.1 Cartesian product ¶
      - depends_on definition 3.43 Map ¶
  - 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 4.4 Internal binary operation; magma ¶ ↺
  - depends_on definition 4.32 Field ¶
    - 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 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.18 Semigroup ¶ ↺
  - depends_on definition 4.31 Module ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.28 Distributivity ¶ ↺
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on definition 4.30 Ring ¶ ↺

Supports

depends_on definition 24.40 Lie–Poisson bracket ¶
- depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
  - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
    - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
  - depends_on remark 24.42 The intermediate axis ¶
- depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
depends_on proposition 5.167 Adjoint representation ¶
- depends_on corollary 5.168 cor:rep-adjoint-faithful ¶
depends_on proposition 5.139 Dimension of the symplectic group ¶
- depends_on proposition 24.5 Properties of the symplectic group ¶
  - depends_on remark 24.6 Connectedness, and what it rests on ¶
  - depends_on remark 24.7 Contrast with the orthogonal group ¶
- depends_on remark 22.21 The count ¶
depends_on remark 22.31 The bracket is a Lie algebra structure ¶
- depends_on remark 22.39 Poisson's theorem does not manufacture new constants ¶

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`	→	Algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5464
`depends_on`	←	Lie–Poisson bracket	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1325
`depends_on`	←	Adjoint representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7822
`depends_on`	←	Dimension of the symplectic group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6107
`depends_on`	←	The bracket is a Lie algebra structure	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1096