theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra

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 15.3 Tensor product Lie algebra ¶ ↺
• depends_on lemma 15.12 Artin decomposition ¶ ↺
• depends_on proposition 15.6 Functoriality ¶
  • 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-2 ¶
• 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 ¶ ↺
    • 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-8 ¶

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