Theorems · Theorem · nonassociative algebras

LieModule.disjoint_lowerCentralSeries_maxTrivSubmodule_iff

∀ (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] [inst_6 : LieModule R L M]
  [LieModule.IsNilpotent L M],
  Disjoint (LieModule.lowerCentralSeries R L M 1) (LieModule.maxTrivSubmodule R L M) ↔ LieModule.IsTrivial L M

For a nilpotent Lie module M of a Lie algebra L, the first term in the lower central series of M contains a non-zero element on which L acts trivially unless the entire action is trivial. Taking M = L, this provides a useful characterisation of Abelian-ness for nilpotent Lie algebras.

Defined in

Mathlib.Algebra.Lie.Nilpotent

Cited by

1 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieModule.IsNilpotent