Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.restrict_killingForm_eq_sum

∀ {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] [LieAlgebra.IsKilling K L]
  [LieModule.IsTriangularizable K (↥H) L] [inst_7 : CharZero K],
  (killingForm K L).restrict H.toSubmodule =
    ∑ α ∈ LieSubalgebra.root, (LieModule.Weight.toLinear K (↥H) L α).smulRight (LieModule.Weight.toLinear K (↥H) L α)

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