Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg_aux

∀ {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] [LieModule.IsTriangularizable K (↥H) L] {α : LieModule.Weight K (↥H) L} {e f : L},
  e ∈ LieAlgebra.rootSpace H ⇑α →
    f ∈ LieAlgebra.rootSpace H (-⇑α) →
      (∀ (h : ↥H), ⁅h, e⁆ = α h • e) →
        ⁅e, f⁆ =
          ((killingForm K L) e) f •
            ↑((LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α))

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

2 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingFieldLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieAlgebra.IsKillingLieModule.IsTriangularizable