definition 5.121 Algebra

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:5238 · p. 157

ground object -- no derivation owed

Rests on

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 3.28 Set ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶ ↺
- 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 4.3 Algebraic structure ¶ ↺
  - depends_on definition 4.20 Monoid ¶
    - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.18 Semigroup ¶
      - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
  - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
    - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.21 Group ¶
      - depends_on definition 4.11 Inverse element ¶ ↺
      - depends_on definition 4.20 Monoid ¶ ↺
    - 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 5.128 Derivation ¶
- 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 ¶
depends_on definition 5.124 Ideal ¶
- depends_on proposition 5.164 The kernel is an ideal ¶
  - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶
depends_on definition 5.127 Lie algebra ¶
- 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 definition 5.122 Linear hull ¶
- depends_on definition 5.124 Ideal ¶ ↺
- depends_on definition 5.123 Subalgebra ¶
depends_on definition 5.123 Subalgebra ¶ ↺
depends_on definition 5.162 Representation of an algebra ¶
- depends_on definition 5.163 Faithful, equivalent, invariant ¶
  - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
  - depends_on proposition 5.164 The kernel is an ideal ¶ ↺
- depends_on proposition 5.167 Adjoint representation ¶ ↺
- depends_on proposition 5.164 The kernel is an ideal ¶ ↺
depends_on proposition 5.125 prop:lin-matrix-algebra ¶
- depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
- depends_on definition 5.162 Representation of an algebra ¶ ↺
- depends_on definition 9.21 Matrix exponential ¶
  - depends_on proposition 9.22 The exponential and its derivative ¶
    - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
      - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
      - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
      - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
      - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      - depends_on theorem 9.26 Structure of the solutions ¶
- depends_on proposition 5.139 Dimension of the symplectic 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`	→	Internal binary operation; magma	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5258
`depends_on`	→	Vector space	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5258
`depends_on`	←	Derivation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5529
`depends_on`	←	Ideal	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5299
`depends_on`	←	Lie algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5464
`depends_on`	←	Linear hull	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5271
`depends_on`	←	Subalgebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5287
`depends_on`	←	Representation of an algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7599
`depends_on`	←	prop:lin-matrix-algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5360