Theorems · Theorem · nonassociative algebras

LieSubmodule.ucs_le_of_normalizer_eq_self

∀ {R : Type u} {L : Type v} {M : Type w} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : LieRingModule L M] {N₁ : LieSubmodule R L M}
  [inst_6 : LieModule R L M], N₁.normalizer = N₁ → ∀ (k : ℕ), LieSubmodule.ucs k ⊥ ≤ N₁

If a Lie module M contains a self-normalizing Lie submodule N, then all terms of the upper central series of M are contained in N. An important instance of this situation arises from a Cartan subalgebra H ⊆ L with the roles of L, M, N played by H, L, H, respectively.

Defined in

Mathlib.Algebra.Lie.Nilpotent

Cited by

3 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModule