isSolvable_of_isNilpotent — Mathlib

English

IsNilpotent of a top Lie subalgebra is equivalent to IsNilpotent of the Lie algebra itself.

Русский

Нильпотентность топового подалгебра эквивалентна нильпотентности самой алгебры.

LaTeX

$$[@simp] isNilpotent_of_top_iff' : IsNilpotent (LieSubalgebra.top) M ↔ IsNilpotent L M$$

Lean4

instance (priority := 100) isSolvable_of_isNilpotent (L : Type v) [LieRing L] [hL : LieModule.IsNilpotent L L] :
    LieAlgebra.IsSolvable L :=
  by
  obtain ⟨k, h⟩ : ∃ k, LieModule.lowerCentralSeries ℤ L L k = ⊥ := hL.nilpotent_int
  use k; rw [← le_bot_iff] at h ⊢
  exact le_trans (LieModule.derivedSeries_le_lowerCentralSeries ℤ L k) h

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