proposition 5.167 Adjoint representation

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:7806 · p. 191

Rests on

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 4.2 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 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
depends_on definition 5.162 Representation of an algebra ¶
- depends_on definition 5.121 Algebra ¶ ↺
- depends_on proposition 5.125 prop:lin-matrix-algebra ¶
  - depends_on definition 5.121 Algebra ¶ ↺
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶ ↺
  - 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 ¶
depends_on equation 5.150 eq:lin-ctesesc2 ¶ ↺
depends_on proposition 5.45 prop:lin-matrix-unique ¶ ↺
proves proof ch:03-linear-algebra-representations@proof-72 ¶

Supports

depends_on corollary 5.168 cor:rep-adjoint-faithful ¶

Neighborhood

Every logical edge within two steps of this node.

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

Edges

type	direction	node	provenance	where
`depends_on`	→	Lie algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7822
`depends_on`	→	Representation of an algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7822
`depends_on`	→	eq:lin-ctesesc2	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7822
`depends_on`	→	prop:lin-matrix-unique	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7822
`depends_on`	←	cor:rep-adjoint-faithful	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7869
`proves`	←	ch:03-linear-algebra-representations@proof-72	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7825