proposition 15.8 The radical of $A$ is solvable in $\mathfrak{g}_{A}$

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:210 · p. 603

Rests on

depends_on definition 15.3 Tensor product Lie algebra ¶
depends_on proposition 15.7 Ideals of $A$ become ideals of $\mathfrak{g}_{A}$ ¶
- depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
- depends_on theorem 15.4 The construction produces a Lie algebra ¶
  - depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
  - proves proof ch:13-lie-algebra-expansions@proof-1 ¶
- proves proof ch:13-lie-algebra-expansions@proof-3 ¶
proves proof ch:13-lie-algebra-expansions@proof-4 ¶

Supports

depends_on proposition 15.15 Takiff brackets ¶
- depends_on example 15.16 The Maxwell tower ¶
- depends_on proposition 15.26 The contraction along a grading ¶
  - depends_on corollary 15.44 The eight vertices are the eight degenerations of $A$ ¶
    - depends_on remark 15.45 A correction this forces: the general splitting gives the static algebra, not Carroll ¶
  - depends_on corollary 15.27 The flat limit ¶
    - depends_on corollary 15.36 Bargmann in $3{+}1$, extended Bargmann in $2{+}1$ ¶
      - depends_on example 15.64 The whole chapter at $D=4$ ¶
      - depends_on remark 15.59 The central charge is what makes the form work ¶
- depends_on proposition 15.21 Resonance ¶
depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶

Neighborhood

Every logical edge within two steps of this node.

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

Edges

type	direction	node	provenance	where
`depends_on`	→	Tensor product Lie algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:218
`depends_on`	→	Ideals of $A$ become ideals of $\mathfrak{g}_{A}$	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:218
`depends_on`	←	Takiff brackets	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:526
`depends_on`	←	Classification of $\mathfrak{g}_{A}$ by the algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:449
`proves`	←	ch:13-lie-algebra-expansions@proof-4	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:221