Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.chainLength_smul

∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : CharZero K] [inst_2 : LieRing L] [inst_3 : LieAlgebra K L]
  [inst_4 : LieAlgebra.IsKilling K L] [inst_5 : FiniteDimensional K L] {H : LieSubalgebra K L}
  [inst_6 : H.IsCartanSubalgebra] [inst_7 : LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L)
  {x : L},
  x ∈ LieAlgebra.rootSpace H ⇑(LieModule.chainTop (⇑α) β) →
    ↑(LieAlgebra.IsKilling.chainLength α β) • x = ⁅LieAlgebra.IsKilling.coroot α, x⁆

Defined in

Mathlib.Algebra.Lie.Weights.RootSystem

Cited by

3 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldCharZeroLieRingLieAlgebraLieAlgebra.IsKillingFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieModule.IsTriangularizable