proposition 15.42 The $\Gamma$-graded commutative algebras

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1564 · p. 618

Rests on

depends_on corollary 15.11 Semisimplicity criterion ¶
- depends_on lemma 15.12 Artin decomposition ¶
  - depends_on notation 15.1 Standing conventions of this chapter ¶
  - depends_on theorem 8.19 Fundamental theorem of algebra ¶
    - depends_on theorem 8.18 Liouville ¶
      - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - proves proof ch:06-complex-analysis@proof-12 ¶
    - proves proof ch:06-complex-analysis@proof-13 ¶
  - proves proof ch:13-lie-algebra-expansions@proof-7 ¶
- depends_on theorem 15.10 Factorisation of the Killing form ¶
  - depends_on definition 15.3 Tensor product Lie algebra ¶
  - depends_on definition 15.9 Trace form of an algebra ¶
  - proves proof ch:13-lie-algebra-expansions@proof-5 ¶
- proves proof ch:13-lie-algebra-expansions@proof-6 ¶
depends_on definition 15.9 Trace form of an algebra ¶ ↺
depends_on lemma 15.40 The three symmetric cosets are one $\Z_{2}\times\Z_{2}$-grading ¶
- 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 ¶
proves proof ch:13-lie-algebra-expansions@proof-25 ¶

Supports

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 ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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declared in the source
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Edges

type	direction	node	provenance	where
`depends_on`	→	Semisimplicity criterion	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1587
`depends_on`	→	Trace form of an algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1587
`depends_on`	→	The three symmetric cosets are one $\Z_{2}\times\Z_{2}$-grading	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1587
`depends_on`	←	The sign data, hypothesis (H3), and the count eleven	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1779
`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-25	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1590