Theorems · Theorem · nonassociative algebras

LieModule.exists_forall_mem_corootSpace_smul_add_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) [inst_7 : H.IsCartanSubalgebra] [inst_8 : IsNoetherian R L] [IsDomain R]
  [IsPrincipalIdealRing R] [CharZero R] [Module.IsTorsionFree R M] [IsNoetherian R M],
  α ≠ 0 → LieModule.genWeightSpace M χ ≠ ⊥ → ∃ a b, 0 < b ∧ ∀ x ∈ LieAlgebra.corootSpace α, (a • α + b • χ) x = 0

Given a (potential) root α relative to a Cartan subalgebra H, if we restrict to the ideal I = corootSpace α of H (informally, I = ⁅H(α), H(-α)⁆), we may find an integral linear combination between α and any weight χ of a representation. This is Proposition 4.4 from [carter2005] and is a key step in the proof that the roots of a semisimple Lie algebra form a root system. It shows that the restriction of α to I vanishes iff the restriction of every root to I vanishes (which cannot happen in a semisimple Lie algebra).

Defined in

Mathlib.Algebra.Lie.Weights.Chain

Cited by

1 results in Mathlib

Foundations

Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieSubalgebra.IsCartanSubalgebraIsNoetherianIsDomainIsPrincipalIdealRingCharZeroModule.IsTorsionFreeIsNoetherian