lemma 15.12 Artin decomposition

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:315 · p. 604

Rests on

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 equation 8.6 eq:cpx-contour-integral ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:06-complex-analysis@proof-5 ¶
  - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
    - depends_on lemma 8.9 ML estimate ¶
      - depends_on equation 8.6 eq:cpx-contour-integral ¶ ↺
      - proves proof ch:06-complex-analysis@proof-4 ¶
    - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
      - depends_on lemma 8.9 ML estimate ¶ ↺
      - proves proof ch:06-complex-analysis@proof-10 ¶
    - proves proof ch:06-complex-analysis@proof-11 ¶
  - 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 ¶

Supports

depends_on corollary 15.11 Semisimplicity criterion ¶
- 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 proposition 15.28 The quadratic family ¶
  - depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶
    - depends_on example 15.64 The whole chapter at $D=4$ ¶
    - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶
    - depends_on remark 15.33 Which branch nature is on ¶
  - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶ ↺
  - depends_on theorem 15.29 The three constant-curvature kinematics, in every dimension ¶
    - depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶ ↺
    - depends_on example 15.64 The whole chapter at $D=4$ ¶ ↺
- depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶
depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶ ↺
depends_on theorem 15.68 Positive Gram forms are positive sums of real characters ¶
- depends_on corollary 15.69 No positive pairing survives the radical ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Standing conventions of this chapter	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:326
`depends_on`	→	Fundamental theorem of algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:326
`depends_on`	←	Semisimplicity criterion	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:295
`depends_on`	←	Classification of $\mathfrak{g}_{A}$ by the algebra	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:449
`depends_on`	←	Positive Gram forms are positive sums of real characters	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2495
`proves`	←	ch:13-lie-algebra-expansions@proof-7	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:329