Theorems · Theorem · nonassociative algebras

LieIdeal.rootSpace_le_of_apply_coroot_ne_zero

∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : LieRing L] [inst_2 : LieAlgebra K L]
  [inst_3 : FiniteDimensional K L] {H : LieSubalgebra K L} [inst_4 : H.IsCartanSubalgebra] [CharZero K]
  [inst_6 : LieAlgebra.IsKilling K L] [LieModule.IsTriangularizable K (↥H) L] (I : LieIdeal K L)
  {α : LieModule.Weight K (↥H) L},
  LieAlgebra.rootSpace H ⇑α ≤ LieSubmodule.restr I H →
    ∀ {γ : ↥H → K}, γ (LieAlgebra.IsKilling.coroot α) ≠ 0 → LieAlgebra.rootSpace H γ ≤ LieSubmodule.restr I H

Defined in

Mathlib.Algebra.Lie.Weights.IsSimple

Cited by

2 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLieRingLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraCharZeroLieAlgebra.IsKillingLieModule.IsTriangularizable