Theorems · Theorem · nonassociative algebras

LieModule.lie_mem_genWeightSpaceChain_of_genWeightSpace_eq_bot_left

∀ {R : Type u_1} {L : Type u_2} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L] (M : Type u_3)
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : LieRingModule L M] [inst_6 : LieModule R L M]
  {H : LieSubalgebra R L} (α χ : ↥H → R) (p q : ℤ) [inst_7 : LieRing.IsNilpotent ↥H],
  LieModule.genWeightSpace M (p • α + χ) = ⊥ →
    ∀ {x : L},
      x ∈ LieAlgebra.rootSpace H (-α) →
        ∀ {y : M}, y ∈ LieModule.genWeightSpaceChain M α χ p q → ⁅x, y⁆ ∈ LieModule.genWeightSpaceChain M α χ p q

Defined in

Mathlib.Algebra.Lie.Weights.Chain

Cited by

1 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieRing.IsNilpotent