theorem 15.4 The construction produces a Lie algebra

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:84 · p. 602

Rests on

depends_on definition 15.3 Tensor product Lie algebra ¶
proves proof ch:13-lie-algebra-expansions@proof-1 ¶

Supports

depends_on proposition 15.6 Functoriality ¶
- depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶
- depends_on theorem 15.34 The truncation tower is a tower of central extensions ¶
  - 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 corollary 15.35 Where the central charges are ¶
    - 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 proposition 15.7 Ideals of $A$ become ideals of $\mathfrak{g}_{A}$ ¶
- depends_on proposition 15.8 The radical of $A$ is solvable in $\mathfrak{g}_{A}$ ¶
  - 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 corollary 15.27 The flat limit ¶
    - depends_on proposition 15.21 Resonance ¶
  - depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶ ↺
depends_on proposition 15.60 Semigroup expansion ¶
- depends_on proposition 15.62 The complex line is not a semigroup algebra ¶
  - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶

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`	→	Tensor product Lie algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:97
`depends_on`	←	Functoriality	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:164
`depends_on`	←	Ideals of $A$ become ideals of $\mathfrak{g}_{A}$	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:195
`depends_on`	←	Semigroup expansion	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2281
`proves`	←	ch:13-lie-algebra-expansions@proof-1	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:100