Theorems · Theorem · nonassociative algebras

LieAlgebra.mem_ker_killingForm_of_mem_rootSpace_of_forall_rootSpace_neg

∀ (K : Type u_2) (L : Type u_3) [inst : LieRing L] [inst_1 : Field K] [inst_2 : LieAlgebra K L] [FiniteDimensional K L]
  (H : LieSubalgebra K L) [inst_4 : H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] {α : ↥H → K} {x : L},
  x ∈ LieAlgebra.rootSpace H α →
    (∀ y ∈ LieAlgebra.rootSpace H (-α), ((killingForm K L) x) y = 0) → x ∈ LinearMap.ker (killingForm K L)

Elements of the α root space which are Killing-orthogonal to the -α root space are Killing-orthogonal to all of L.

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

3 results in Mathlib

Foundations

Depth 128 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingFieldLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieModule.IsTriangularizable