Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.cartanEquivDual_symm_apply_mem_corootSpace

∀ {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]
  [inst_5 : LieAlgebra.IsKilling K L] [LieModule.IsTriangularizable K (↥H) L] (α : LieModule.Weight K (↥H) L),
  (LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α) ∈ LieAlgebra.corootSpace ⇑α

This is Proposition 4.18 from [carter2005] except that we use LieModule.exists_forall_lie_eq_smul instead of Lie's theorem (and so avoid assuming K has characteristic zero).

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

3 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingFieldLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieAlgebra.IsKillingLieModule.IsTriangularizable