Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.mem_sl2SubalgebraOfRoot_iff

∀ {K : Type u_2} {L : Type u_3} [inst : LieRing L] [inst_1 : Field K] [inst_2 : LieAlgebra K L]
  [inst_3 : FiniteDimensional K L] {H : LieSubalgebra K L} [inst_4 : H.IsCartanSubalgebra]
  [inst_5 : LieAlgebra.IsKilling K L] [inst_6 : LieModule.IsTriangularizable K (↥H) L] [inst_7 : CharZero K]
  {α : LieModule.Weight K (↥H) L} (hα : α.IsNonZero) {h e f : L},
  IsSl2Triple h e f →
    e ∈ LieAlgebra.rootSpace H ⇑α →
      f ∈ LieAlgebra.rootSpace H (-⇑α) →
        ∀ {x : L}, x ∈ LieAlgebra.IsKilling.sl2SubalgebraOfRoot hα ↔ ∃ c₁ c₂ c₃, x = c₁ • e + c₂ • f + c₃ • ⁅e, f⁆

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

1 results in Mathlib

Foundations

Depth 191 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingFieldLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieAlgebra.IsKillingLieModule.IsTriangularizableCharZero