Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.rootSpace_zsmul_add_ne_bot_iff_mem

∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : CharZero K] [inst_2 : LieRing L] [inst_3 : LieAlgebra K L]
  [LieAlgebra.IsKilling K L] [inst_5 : FiniteDimensional K L] {H : LieSubalgebra K L} [inst_6 : H.IsCartanSubalgebra]
  [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L),
  α.IsNonZero →
    ∀ (n : ℤ),
      LieAlgebra.rootSpace H (n • ⇑α + ⇑β) ≠ ⊥ ↔
        n ∈ Finset.Icc (-↑(LieModule.chainBotCoeff (⇑α) β)) ↑(LieModule.chainTopCoeff (⇑α) β)

Defined in

Mathlib.Algebra.Lie.Weights.RootSystem

Cited by

1 results in Mathlib

Foundations

Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldCharZeroLieRingLieAlgebraLieAlgebra.IsKillingFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieModule.IsTriangularizable