Theorems · Theorem · nonassociative algebras

LieModule.trace_toEnd_genWeightSpaceChain_eq_zero

∀ {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 : H.IsCartanSubalgebra] [inst_8 : IsNoetherian R L],
  LieModule.genWeightSpace M (p • α + χ) = ⊥ →
    LieModule.genWeightSpace M (q • α + χ) = ⊥ →
      ∀ {x : ↥H},
        x ∈ LieAlgebra.corootSpace α →
          (LinearMap.trace R ↥(LieModule.genWeightSpaceChain M α χ p q))
              ((LieModule.toEnd R ↥H ↥(LieModule.genWeightSpaceChain M α χ p q)) x) =
            0

Defined in

Mathlib.Algebra.Lie.Weights.Chain

Cited by

1 results in Mathlib

Foundations

Depth 108 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieSubalgebra.IsCartanSubalgebraIsNoetherian