Theorems · Theorem · nonassociative algebras

LieAlgebra.Basis.root_mem_or_mem_neg

∀ {ι : Type u_1} {K : Type u_2} {L : Type u_3} [inst : Fintype ι] [inst_1 : Field K] [inst_2 : CharZero K]
  [inst_3 : LieRing L] [inst_4 : LieAlgebra K L] [inst_5 : FiniteDimensional K L] {H : LieSubalgebra K L}
  (b : LieAlgebra.Basis ι H) [inst_6 : LieModule.IsTriangularizable K (↥H) L] [inst_7 : LieAlgebra.IsKilling K L]
  (χ : ↥LieSubalgebra.root),
  (LieAlgebra.IsKilling.rootSystem H).root χ ∈
      AddSubmonoid.closure (⇑(LieAlgebra.IsKilling.rootSystem H).root '' Set.range b.baseSupp') ∨
    -(LieAlgebra.IsKilling.rootSystem H).root χ ∈
      AddSubmonoid.closure (⇑(LieAlgebra.IsKilling.rootSystem H).root '' Set.range b.baseSupp')

Defined in

Mathlib.Algebra.Lie.Basis.Base

Cited by

0 results in Mathlib

Foundations

Depth 207 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeFieldCharZeroLieRingLieAlgebraFiniteDimensionalLieModule.IsTriangularizableLieAlgebra.IsKilling