proposition 15.26 The contraction along a grading

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:781 · p. 609

Rests on

depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶
depends_on definition 15.20 Resonant subalgebra ¶
- depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶ ↺
- depends_on definition 15.14 Truncated polynomial algebra and Takiff algebra ¶
  - depends_on definition 15.3 Tensor product Lie algebra ¶
depends_on equation 14.125 eq:lie-contraction ¶
depends_on proposition 15.15 Takiff brackets ¶
- depends_on definition 15.14 Truncated polynomial algebra and Takiff algebra ¶ ↺
- depends_on definition 15.3 Tensor product Lie algebra ¶ ↺
- depends_on proposition 15.8 The radical of $A$ is solvable in $\mathfrak{g}_{A}$ ¶
  - 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 ¶
- proves proof ch:13-lie-algebra-expansions@proof-9 ¶
proves proof ch:13-lie-algebra-expansions@proof-14 ¶

Supports

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 ¶

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`	→	$\Z_{2}$-grading, symmetric coset	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:794
`depends_on`	→	Resonant subalgebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:794
`depends_on`	→	eq:lie-contraction	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:794
`depends_on`	→	Takiff brackets	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:794
`depends_on`	←	The eight vertices are the eight degenerations of $A$	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1713
`depends_on`	←	The flat limit	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:836
`proves`	←	ch:13-lie-algebra-expansions@proof-14	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:797