Theorems · Theorem · nonassociative algebras

LieModule.nilpotentOfNilpotentQuotient

∀ (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]
  {N : LieSubmodule R L M},
  N ≤ LieModule.maxTrivSubmodule R L M → LieModule.IsNilpotent L (M ⧸ N) → LieModule.IsNilpotent L M

If the quotient of a Lie module M by a Lie submodule on which the Lie algebra acts trivially is nilpotent then M is nilpotent. This is essentially the Lie module equivalent of the fact that a central extension of nilpotent Lie algebras is nilpotent. See LieAlgebra.nilpotent_of_nilpotent_quotient below for the corresponding result for Lie algebras.

Defined in

Mathlib.Algebra.Lie.Nilpotent

Cited by

1 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModule