Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.eq_zero_of_apply_eq_zero_of_mem_corootSpace

∀ {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] [CharZero K] (x : ↥H) (α : ↥H → K),
  α x = 0 → x ∈ LieAlgebra.corootSpace α → x = 0

The contrapositive of this result is very useful, taking x to be the element of H corresponding to a root α under the identification between H and H^* provided by the Killing form.

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

1 results in Mathlib

Foundations

Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingFieldLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieAlgebra.IsKillingLieModule.IsTriangularizableCharZero