Theorems · Theorem · nonassociative algebras

LieAlgebra.mem_biSup_genWeightSpace_of

∀ {R : Type u_1} {L : Type u_2} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
  {H : LieSubalgebra R L} [inst_3 : LieRing.IsNilpotent ↥H] {M : Type u_3} [inst_4 : AddCommGroup M]
  [inst_5 : Module R M] [inst_6 : LieRingModule L M] [inst_7 : LieModule R L M] {s : Set (↥H → R)},
  (∀ χ₁ ∈ s, ∀ χ₂ ∈ s, χ₁ + χ₂ ∈ s) →
    ∀ {x : L} {m : M},
      x ∈ ⨆ χ ∈ s, LieAlgebra.rootSpace H χ →
        m ∈ ⨆ χ ∈ s, LieModule.genWeightSpace M χ → ⁅x, m⁆ ∈ ⨆ χ ∈ s, LieModule.genWeightSpace M χ

Defined in

Mathlib.Algebra.Lie.Weights.Cartan

Cited by

1 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraLieRing.IsNilpotentAddCommGroupModuleLieRingModuleLieModule