lemma 15.40 The three symmetric cosets are one $\Z_{2}\times\Z_{2}$-grading

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1465 · p. 617

Rests on

depends_on definition 15.24 The four kinematical gradings ¶
- depends_on notation 15.22 Generators of $\mathfrak{so}(D-1,2)$ ¶
  - depends_on equation 14.90 eq:lie-algebrasopq ¶
depends_on lemma 15.19 A grading is an involution, and the Killing form respects it ¶
- depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶
- proves proof ch:13-lie-algebra-expansions@proof-10 ¶
depends_on proposition 15.23 Brackets in the kinematical splitting ¶
- depends_on equation 14.90 eq:lie-algebrasopq ¶ ↺
- depends_on equation 14.128 eq:lie-kinematical-constants ¶
- depends_on notation 15.22 Generators of $\mathfrak{so}(D-1,2)$ ¶ ↺
- proves proof ch:13-lie-algebra-expansions@proof-12 ¶
depends_on proposition 15.25 Which of the four are symmetric cosets ¶
- depends_on definition 15.24 The four kinematical gradings ¶ ↺
- depends_on definition 15.18 $\Z_{2}$-grading, symmetric coset ¶ ↺
- depends_on proposition 15.23 Brackets in the kinematical splitting ¶ ↺
- proves proof ch:13-lie-algebra-expansions@proof-13 ¶
proves proof ch:13-lie-algebra-expansions@proof-23 ¶

Supports

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 $A$ ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	The four kinematical gradings	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1487
`depends_on`	→	A grading is an involution, and the Killing form respects it	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1487
`depends_on`	→	Brackets in the kinematical splitting	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1487
`depends_on`	→	Which of the four are symmetric cosets	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1487
`depends_on`	←	The $\Gamma$-graded commutative algebras	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1587
`depends_on`	←	The three constants of the classification are the three structure constants of $A$	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1644
`proves`	←	ch:13-lie-algebra-expansions@proof-23	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1490