Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.restr_invtSubmoduleToLieIdeal_eq_iSup

∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : CharZero K] [inst_2 : LieRing L] [inst_3 : LieAlgebra K L]
  [inst_4 : FiniteDimensional K L] [inst_5 : LieAlgebra.IsKilling K L] {H : LieSubalgebra K L}
  [inst_6 : H.IsCartanSubalgebra] [inst_7 : LieModule.IsTriangularizable K (↥H) L] (q : Submodule K (Module.Dual K ↥H))
  (hq : ∀ (i : ↥LieSubalgebra.root), q ∈ Module.End.invtSubmodule ↑((LieAlgebra.IsKilling.rootSystem H).reflection i)),
  LieSubmodule.restr (LieAlgebra.IsKilling.invtSubmoduleToLieIdeal q hq) H =
    ⨆ α, LieAlgebra.IsKilling.sl2SubmoduleOfRoot ⋯

Defined in

Mathlib.Algebra.Lie.Weights.IsSimple

Cited by

1 results in Mathlib

Foundations

Depth 213 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldCharZeroLieRingLieAlgebraFiniteDimensionalLieAlgebra.IsKillingLieSubalgebra.IsCartanSubalgebraLieModule.IsTriangularizable