Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.traceForm_coroot

∀ {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] (α : LieModule.Weight K (↥H) L) (x : ↥H),
  ((LieModule.traceForm K (↥H) L) (LieAlgebra.IsKilling.coroot α)) x =
    2 • (α ((LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α)))⁻¹ • α x

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

1 results in Mathlib

Foundations

Depth 187 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingFieldLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieAlgebra.IsKilling