Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.lieIdeal_eq_inf_cartan_sup_biSup_rootSpace

∀ {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] (I : LieIdeal K L),
  LieSubmodule.restr I H =
    LieSubmodule.restr I H ⊓ H.toLieSubmodule ⊔
      ⨆ α, ⨆ (_ : LieAlgebra.rootSpace H ⇑↑α ≤ LieSubmodule.restr I H), LieAlgebra.rootSpace H ⇑↑α

In a Lie algebra with non-degenerate Killing form, a Lie ideal decomposes as its intersection with the Cartan subalgebra plus a sum of root spaces corresponding to some subset of roots.

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

1 results in Mathlib

Foundations

Depth 191 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingFieldLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieAlgebra.IsKillingLieModule.IsTriangularizableCharZero