Theorems · Theorem · nonassociative algebras

LieModule.exists_forall_lie_eq_smul

∀ (R : Type u_2) (L : Type u_3) (M : Type u_4) [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]
  [inst_7 : LieRing.IsNilpotent L] [LieModule.LinearWeights R L M] [IsNoetherian R M] (χ : LieModule.Weight R L M),
  ∃ m, m ≠ 0 ∧ ∀ (x : L), ⁅x, m⁆ = χ x • m

Given a Lie module M of a nilpotent Lie algebra L with coefficients in R, if a function χ : L → R has a simultaneous generalized eigenvector for the action of L then it has a simultaneous true eigenvector, provided M is Noetherian and has linear weights.

Defined in

Mathlib.Algebra.Lie.Weights.Linear

Cited by

2 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieRing.IsNilpotentLieModule.LinearWeightsIsNoetherian