lemma 15.19 A grading is an involution, and the Killing form respects it

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:591 · p. 607

Rests on

depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶
proves proof ch:13-lie-algebra-expansions@proof-10 ¶

Supports

depends_on lemma 15.40 The three symmetric cosets are one $\Z_{2}\times\Z_{2}$-grading ¶
- depends_on proposition 15.42 The $\Gamma$-graded commutative algebras ¶
  - depends_on corollary 15.46 The sign data, hypothesis (H3), and the count eleven ¶
  - depends_on theorem 15.43 The three constants of the classification are the three structure constants of $A$ ¶
    - depends_on corollary 15.46 The sign data, hypothesis (H3), and the count eleven ¶ ↺
    - 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 theorem 15.43 The three constants of the classification are the three structure constants of $A$ ¶ ↺
depends_on theorem 15.58 Nondegeneracy on a resonant subalgebra ¶
- depends_on corollary 15.69 No positive pairing survives the radical ¶
- 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:601
`depends_on`	←	The three symmetric cosets are one $\Z_{2}\times\Z_{2}$-grading	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1487
`depends_on`	←	Nondegeneracy on a resonant subalgebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2200
`proves`	←	ch:13-lie-algebra-expansions@proof-10	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:604