Theorems · Theorem · nonassociative algebras

LieAlgebra.lie_mem_genWeightSpace_of_mem_genWeightSpace

∀ {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] {χ₁ χ₂ : ↥H → R} {x : L} {m : M},
  x ∈ LieAlgebra.rootSpace H χ₁ → m ∈ LieModule.genWeightSpace M χ₂ → ⁅x, m⁆ ∈ LieModule.genWeightSpace M (χ₁ + χ₂)

Defined in

Mathlib.Algebra.Lie.Weights.Cartan

Cited by

3 results in Mathlib

Foundations

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

Assumes

CommRingLieRingLieAlgebraLieRing.IsNilpotentAddCommGroupModuleLieRingModuleLieModule